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Theorem fpwrelmap 31650
Description: Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 10384 and marypha2lem1 9371. (Contributed by Thierry Arnoux, 28-Aug-2017.)
Hypotheses
Ref Expression
fpwrelmap.1 𝐴 ∈ V
fpwrelmap.2 𝐵 ∈ V
fpwrelmap.3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
Assertion
Ref Expression
fpwrelmap 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝐵,𝑓,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem fpwrelmap
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 fpwrelmap.3 . . 3 𝑀 = (𝑓 ∈ (𝒫 𝐵m 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
2 fpwrelmap.1 . . . . . 6 𝐴 ∈ V
32a1i 11 . . . . 5 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝐴 ∈ V)
4 simpr 485 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦 ∈ (𝑓𝑥))
5 elmapi 8787 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
65ffvelcdmda 7035 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
76adantr 481 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑓𝑥) ∈ 𝒫 𝐵)
8 elelpwi 4570 . . . . . . . . 9 ((𝑦 ∈ (𝑓𝑥) ∧ (𝑓𝑥) ∈ 𝒫 𝐵) → 𝑦𝐵)
94, 7, 8syl2anc 584 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
109ex 413 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
1110alrimiv 1930 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
12 abss 4017 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
13 fpwrelmap.2 . . . . . . . 8 𝐵 ∈ V
1413ssex 5278 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1512, 14sylbir 234 . . . . . 6 (∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1611, 15syl 17 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
173, 16opabex3d 7898 . . . 4 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
1817adantl 482 . . 3 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵m 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
192mptex 7173 . . . 4 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V
2019a1i 11 . . 3 ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V)
2110imdistanda 572 . . . . . . . . . 10 (𝑓 ∈ (𝒫 𝐵m 𝐴) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
2221ssopab2dv 5508 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵m 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2322adantr 481 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
24 simpr 485 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
25 df-xp 5639 . . . . . . . . 9 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
2625a1i 11 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2723, 24, 263sstr4d 3991 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28 velpw 4565 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
2927, 28sylibr 233 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
305feqmptd 6910 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵m 𝐴) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
3130adantr 481 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
32 nfv 1917 . . . . . . . . 9 𝑥 𝑓 ∈ (𝒫 𝐵m 𝐴)
33 nfopab1 5175 . . . . . . . . . 10 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3433nfeq2 2924 . . . . . . . . 9 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3532, 34nfan 1902 . . . . . . . 8 𝑥(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
36 df-rab 3408 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)}
3736a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
38 nfv 1917 . . . . . . . . . . . 12 𝑦 𝑓 ∈ (𝒫 𝐵m 𝐴)
39 nfopab2 5176 . . . . . . . . . . . . 13 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4039nfeq2 2924 . . . . . . . . . . . 12 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4138, 40nfan 1902 . . . . . . . . . . 11 𝑦(𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
42 nfv 1917 . . . . . . . . . . 11 𝑦 𝑥𝐴
4341, 42nfan 1902 . . . . . . . . . 10 𝑦((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴)
449adantllr 717 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
45 df-br 5106 . . . . . . . . . . . . . . . . 17 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46 eleq2 2826 . . . . . . . . . . . . . . . . . 18 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
47 opabidw 5481 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
4846, 47bitrdi 286 . . . . . . . . . . . . . . . . 17 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
4945, 48bitrid 282 . . . . . . . . . . . . . . . 16 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5049ad2antlr 725 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
51 elfvdm 6879 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑓𝑥) → 𝑥 ∈ dom 𝑓)
5251adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥 ∈ dom 𝑓)
535fdmd 6679 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝒫 𝐵m 𝐴) → dom 𝑓 = 𝐴)
5453adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → dom 𝑓 = 𝐴)
5552, 54eleqtrd 2840 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝐴)
5655ex 413 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑥𝐴))
5756pm4.71rd 563 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (𝒫 𝐵m 𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5857ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5950, 58bitr4d 281 . . . . . . . . . . . . . 14 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6059biimpar 478 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
6144, 60jca 512 . . . . . . . . . . . 12 ((((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑦𝐵𝑥𝑟𝑦))
6261ex 413 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → (𝑦𝐵𝑥𝑟𝑦)))
6359biimpd 228 . . . . . . . . . . . 12 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6463adantld 491 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → ((𝑦𝐵𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥)))
6562, 64impbid 211 . . . . . . . . . 10 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
6643, 65abbid 2807 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
67 abid2 2875 . . . . . . . . . 10 {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥)
6867a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥))
6937, 66, 683eqtr2rd 2783 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
7035, 69mpteq2da 5203 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7131, 70eqtrd 2776 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7229, 71jca 512 . . . . 5 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
73 ssrab2 4037 . . . . . . . . . . . 12 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
7413, 73elpwi2 5303 . . . . . . . . . . 11 {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵
7574a1i 11 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
7675fmpttd 7063 . . . . . . . . 9 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
7776adantr 481 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
78 simpr 485 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7978feq1d 6653 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
8077, 79mpbird 256 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
8113pwex 5335 . . . . . . . 8 𝒫 𝐵 ∈ V
8281, 2elmap 8809 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8380, 82sylibr 233 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵m 𝐴))
84 elpwi 4567 . . . . . . . . . 10 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
8584adantr 481 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
86 xpss 5649 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
8785, 86sstrdi 3956 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
88 df-rel 5640 . . . . . . . 8 (Rel 𝑟𝑟 ⊆ (V × V))
8987, 88sylibr 233 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
90 relopabv 5777 . . . . . . . 8 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9190a1i 11 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
92 id 22 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
93 nfv 1917 . . . . . . . . 9 𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
94 nfmpt1 5213 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9594nfeq2 2924 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9693, 95nfan 1902 . . . . . . . 8 𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
97 nfv 1917 . . . . . . . . 9 𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
9842nfci 2890 . . . . . . . . . . 11 𝑦𝐴
99 nfrab1 3426 . . . . . . . . . . 11 𝑦{𝑦𝐵𝑥𝑟𝑦}
10098, 99nfmpt 5212 . . . . . . . . . 10 𝑦(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
101100nfeq2 2924 . . . . . . . . 9 𝑦 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
10297, 101nfan 1902 . . . . . . . 8 𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
103 nfcv 2907 . . . . . . . 8 𝑥𝑟
104 nfcv 2907 . . . . . . . 8 𝑦𝑟
105 brelg 31528 . . . . . . . . . . . . . . . 16 ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
10684, 105sylan 580 . . . . . . . . . . . . . . 15 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
107106adantlr 713 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
108107simpld 495 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝐴)
109107simprd 496 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦𝐵)
110 simpr 485 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
11178fveq1d 6844 . . . . . . . . . . . . . . . . . 18 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓𝑥) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥))
11213rabex 5289 . . . . . . . . . . . . . . . . . . 19 {𝑦𝐵𝑥𝑟𝑦} ∈ V
113 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
114113fvmpt2 6959 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ {𝑦𝐵𝑥𝑟𝑦} ∈ V) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
115112, 114mpan2 689 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
116111, 115sylan9eq 2796 . . . . . . . . . . . . . . . . 17 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
117116eleq2d 2823 . . . . . . . . . . . . . . . 16 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦}))
118 rabid 3427 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦} ↔ (𝑦𝐵𝑥𝑟𝑦))
119117, 118bitrdi 286 . . . . . . . . . . . . . . 15 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
120108, 119syldan 591 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
121109, 110, 120mpbir2and 711 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥))
122108, 121jca 512 . . . . . . . . . . . 12 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
123122ex 413 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
124119simplbda 500 . . . . . . . . . . . 12 ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
125124expl 458 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦))
126123, 125impbid 211 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
12745, 126bitr3id 284 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
128127, 47bitr4di 288 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
12996, 102, 103, 104, 33, 39, 128eqrelrd2 31535 . . . . . . 7 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13089, 91, 92, 129syl21anc 836 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13183, 130jca 512 . . . . 5 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13272, 131impbii 208 . . . 4 ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
133132a1i 11 . . 3 (⊤ → ((𝑓 ∈ (𝒫 𝐵m 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))))
1341, 18, 20, 133f1od 7605 . 2 (⊤ → 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
135134mptru 1548 1 𝑀:(𝒫 𝐵m 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wtru 1542  wcel 2106  {cab 2713  {crab 3407  Vcvv 3445  wss 3910  𝒫 cpw 4560  cop 4592   class class class wbr 5105  {copab 5167  cmpt 5188   × cxp 5631  dom cdm 5633  Rel wrel 5638  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  m cmap 8765
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767
This theorem is referenced by:  fpwrelmapffs  31651
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