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Theorem rngoisocnv 35438
 Description: The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
Assertion
Ref Expression
rngoisocnv ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))

Proof of Theorem rngoisocnv
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1ocnv 6603 . . . . . . . 8 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))
2 f1of 6591 . . . . . . . 8 (𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
31, 2syl 17 . . . . . . 7 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
43ad2antll 728 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹:ran (1st𝑆)⟶ran (1st𝑅))
5 eqid 2798 . . . . . . . . . 10 (2nd𝑅) = (2nd𝑅)
6 eqid 2798 . . . . . . . . . 10 (GId‘(2nd𝑅)) = (GId‘(2nd𝑅))
7 eqid 2798 . . . . . . . . . 10 (2nd𝑆) = (2nd𝑆)
8 eqid 2798 . . . . . . . . . 10 (GId‘(2nd𝑆)) = (GId‘(2nd𝑆))
95, 6, 7, 8rngohom1 35425 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1093expa 1115 . . . . . . . 8 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
1110adantrr 716 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)))
12 eqid 2798 . . . . . . . . . . 11 ran (1st𝑅) = ran (1st𝑅)
1312, 5, 6rngo1cl 35396 . . . . . . . . . 10 (𝑅 ∈ RingOps → (GId‘(2nd𝑅)) ∈ ran (1st𝑅))
14 f1ocnvfv 7014 . . . . . . . . . 10 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (GId‘(2nd𝑅)) ∈ ran (1st𝑅)) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1513, 14sylan2 595 . . . . . . . . 9 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑅 ∈ RingOps) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1615ancoms 462 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1716ad2ant2rl 748 . . . . . . 7 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝐹‘(GId‘(2nd𝑅))) = (GId‘(2nd𝑆)) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅))))
1811, 17mpd 15 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)))
19 f1ocnvfv2 7013 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑥)) = 𝑥)
20 f1ocnvfv2 7013 . . . . . . . . . . . . . 14 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘(𝐹𝑦)) = 𝑦)
2119, 20anim12dan 621 . . . . . . . . . . . . 13 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦))
22 oveq12 7145 . . . . . . . . . . . . 13 (((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2321, 22syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2423adantll 713 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
2524adantll 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(1st𝑆)𝑦))
26 f1ocnvdm 7020 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑥 ∈ ran (1st𝑆)) → (𝐹𝑥) ∈ ran (1st𝑅))
27 f1ocnvdm 7020 . . . . . . . . . . . . . . . 16 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹𝑦) ∈ ran (1st𝑅))
2826, 27anim12dan 621 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)))
29 eqid 2798 . . . . . . . . . . . . . . . 16 (1st𝑅) = (1st𝑅)
30 eqid 2798 . . . . . . . . . . . . . . . 16 (1st𝑆) = (1st𝑆)
3129, 12, 30rngohomadd 35426 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
3228, 31sylan2 595 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
3332exp32 424 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))))
34333expa 1115 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))))
3534impr 458 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦)))))
3635imp 410 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(1st𝑆)(𝐹‘(𝐹𝑦))))
37 eqid 2798 . . . . . . . . . . . . . . . 16 ran (1st𝑆) = ran (1st𝑆)
3830, 37rngogcl 35369 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
39383expb 1117 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆))
40 f1ocnvfv2 7013 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4140ancoms 462 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4239, 41sylan 583 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4342an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4443adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4544adantlrl 719 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝑥(1st𝑆)𝑦))
4625, 36, 453eqtr4rd 2844 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
47 f1of1 6590 . . . . . . . . . . . 12 (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → 𝐹:ran (1st𝑅)–1-1→ran (1st𝑆))
4847ad2antlr 726 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → 𝐹:ran (1st𝑅)–1-1→ran (1st𝑆))
49 f1ocnvdm 7020 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5049ancoms 462 . . . . . . . . . . . . . 14 (((𝑥(1st𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5139, 50sylan 583 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5251an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5352adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅))
5429, 12rngogcl 35369 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
55543expb 1117 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5628, 55sylan2 595 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5756anassrs 471 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
5857adantllr 718 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
59 f1fveq 6999 . . . . . . . . . . 11 ((𝐹:ran (1st𝑅)–1-1→ran (1st𝑆) ∧ ((𝐹‘(𝑥(1st𝑆)𝑦)) ∈ ran (1st𝑅) ∧ ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6048, 53, 58, 59syl12anc 835 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6160adantlrl 719 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(1st𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(1st𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦))))
6246, 61mpbid 235 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)))
63 oveq12 7145 . . . . . . . . . . . . 13 (((𝐹‘(𝐹𝑥)) = 𝑥 ∧ (𝐹‘(𝐹𝑦)) = 𝑦) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6421, 63syl 17 . . . . . . . . . . . 12 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6564adantll 713 . . . . . . . . . . 11 (((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6665adantll 713 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))) = (𝑥(2nd𝑆)𝑦))
6729, 12, 5, 7rngohommul 35427 . . . . . . . . . . . . . . 15 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
6828, 67sylan2 595 . . . . . . . . . . . . . 14 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
6968exp32 424 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))))
70693expa 1115 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))))
7170impr 458 . . . . . . . . . . 11 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ((𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦)))))
7271imp 410 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) = ((𝐹‘(𝐹𝑥))(2nd𝑆)(𝐹‘(𝐹𝑦))))
7330, 7, 37rngocl 35358 . . . . . . . . . . . . . . 15 ((𝑆 ∈ RingOps ∧ 𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
74733expb 1117 . . . . . . . . . . . . . 14 ((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆))
75 f1ocnvfv2 7013 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7675ancoms 462 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7774, 76sylan 583 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7877an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
7978adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
8079adantlrl 719 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝑥(2nd𝑆)𝑦))
8166, 72, 803eqtr4rd 2844 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
82 f1ocnvdm 7020 . . . . . . . . . . . . . . 15 ((𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8382ancoms 462 . . . . . . . . . . . . . 14 (((𝑥(2nd𝑆)𝑦) ∈ ran (1st𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8474, 83sylan 583 . . . . . . . . . . . . 13 (((𝑆 ∈ RingOps ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8584an32s 651 . . . . . . . . . . . 12 (((𝑆 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8685adantlll 717 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅))
8729, 5, 12rngocl 35358 . . . . . . . . . . . . . . 15 ((𝑅 ∈ RingOps ∧ (𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅)) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
88873expb 1117 . . . . . . . . . . . . . 14 ((𝑅 ∈ RingOps ∧ ((𝐹𝑥) ∈ ran (1st𝑅) ∧ (𝐹𝑦) ∈ ran (1st𝑅))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
8928, 88sylan2 595 . . . . . . . . . . . . 13 ((𝑅 ∈ RingOps ∧ (𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆)))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9089anassrs 471 . . . . . . . . . . . 12 (((𝑅 ∈ RingOps ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
9190adantllr 718 . . . . . . . . . . 11 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))
92 f1fveq 6999 . . . . . . . . . . 11 ((𝐹:ran (1st𝑅)–1-1→ran (1st𝑆) ∧ ((𝐹‘(𝑥(2nd𝑆)𝑦)) ∈ ran (1st𝑅) ∧ ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)) ∈ ran (1st𝑅))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9348, 86, 91, 92syl12anc 835 . . . . . . . . . 10 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9493adantlrl 719 . . . . . . . . 9 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝐹‘(𝑥(2nd𝑆)𝑦))) = (𝐹‘((𝐹𝑥)(2nd𝑅)(𝐹𝑦))) ↔ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9581, 94mpbid 235 . . . . . . . 8 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦)))
9662, 95jca 515 . . . . . . 7 ((((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) ∧ (𝑥 ∈ ran (1st𝑆) ∧ 𝑦 ∈ ran (1st𝑆))) → ((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9796ralrimivva 3156 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))
9830, 7, 37, 8, 29, 5, 12, 6isrngohom 35422 . . . . . . . 8 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
9998ancoms 462 . . . . . . 7 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
10099adantr 484 . . . . . 6 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹 ∈ (𝑆 RngHom 𝑅) ↔ (𝐹:ran (1st𝑆)⟶ran (1st𝑅) ∧ (𝐹‘(GId‘(2nd𝑆))) = (GId‘(2nd𝑅)) ∧ ∀𝑥 ∈ ran (1st𝑆)∀𝑦 ∈ ran (1st𝑆)((𝐹‘(𝑥(1st𝑆)𝑦)) = ((𝐹𝑥)(1st𝑅)(𝐹𝑦)) ∧ (𝐹‘(𝑥(2nd𝑆)𝑦)) = ((𝐹𝑥)(2nd𝑅)(𝐹𝑦))))))
1014, 18, 97, 100mpbir3and 1339 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹 ∈ (𝑆 RngHom 𝑅))
1021ad2antll 728 . . . . 5 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))
103101, 102jca 515 . . . 4 (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))) → (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅)))
104103ex 416 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → ((𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆)) → (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
10529, 12, 30, 37isrngoiso 35435 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:ran (1st𝑅)–1-1-onto→ran (1st𝑆))))
10630, 37, 29, 12isrngoiso 35435 . . . 4 ((𝑆 ∈ RingOps ∧ 𝑅 ∈ RingOps) → (𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
107106ancoms 462 . . 3 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑆 RngIso 𝑅) ↔ (𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝐹:ran (1st𝑆)–1-1-onto→ran (1st𝑅))))
108104, 105, 1073imtr4d 297 . 2 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑆 RngIso 𝑅)))
1091083impia 1114 1 ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ◡ccnv 5519  ran crn 5521  ⟶wf 6321  –1-1→wf1 6322  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136  1st c1st 7672  2nd c2nd 7673  GIdcgi 28283  RingOpscrngo 35351   RngHom crnghom 35417   RngIso crngiso 35418 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394  df-grpo 28286  df-gid 28287  df-ablo 28338  df-ass 35300  df-exid 35302  df-mgmOLD 35306  df-sgrOLD 35318  df-mndo 35324  df-rngo 35352  df-rngohom 35420  df-rngoiso 35433 This theorem is referenced by:  riscer  35445
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