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Theorem fpwrelmap 28689
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 9130 and marypha2lem1 8201. (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 475 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦 ∈ (𝑓𝑥))
5 elmapi 7742 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓:𝐴⟶𝒫 𝐵)
65ffvelrnda 6251 . . . . . . . . . 10 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝑓𝑥) ∈ 𝒫 𝐵)
76adantr 479 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑓𝑥) ∈ 𝒫 𝐵)
8 elelpwi 4118 . . . . . . . . 9 ((𝑦 ∈ (𝑓𝑥) ∧ (𝑓𝑥) ∈ 𝒫 𝐵) → 𝑦𝐵)
94, 7, 8syl2anc 690 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
109ex 448 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
1110alrimiv 1841 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
12 abss 3633 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵))
13 fpwrelmap.2 . . . . . . . 8 𝐵 ∈ V
1413ssex 4724 . . . . . . 7 ({𝑦𝑦 ∈ (𝑓𝑥)} ⊆ 𝐵 → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1512, 14sylbir 223 . . . . . 6 (∀𝑦(𝑦 ∈ (𝑓𝑥) → 𝑦𝐵) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
1611, 15syl 17 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} ∈ V)
173, 16opabex3d 7014 . . . 4 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
1817adantl 480 . . 3 ((⊤ ∧ 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ∈ V)
192mptex 6367 . . . 4 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V
2019a1i 11 . . 3 ((⊤ ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) ∈ V)
2110imdistanda 724 . . . . . . . . . 10 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → (𝑥𝐴𝑦𝐵)))
2221ssopab2dv 4918 . . . . . . . . 9 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
2322adantr 479 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)})
24 simpr 475 . . . . . . . 8 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
25 df-xp 5033 . . . . . . . . 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 3610 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵))
28 selpw 4114 . . . . . . 7 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵))
2927, 28sylibr 222 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵))
305feqmptd 6143 . . . . . . . 8 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
3130adantr 479 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ (𝑓𝑥)))
32 nfv 1829 . . . . . . . . 9 𝑥 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)
33 nfopab1 4645 . . . . . . . . . 10 𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3433nfeq2 2765 . . . . . . . . 9 𝑥 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
3532, 34nfan 1815 . . . . . . . 8 𝑥(𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
36 df-rab 2904 . . . . . . . . . 10 {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)}
3736a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
38 nfv 1829 . . . . . . . . . . . 12 𝑦 𝑓 ∈ (𝒫 𝐵𝑚 𝐴)
39 nfopab2 4646 . . . . . . . . . . . . 13 𝑦{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4039nfeq2 2765 . . . . . . . . . . . 12 𝑦 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
4138, 40nfan 1815 . . . . . . . . . . 11 𝑦(𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
42 nfv 1829 . . . . . . . . . . 11 𝑦 𝑥𝐴
4341, 42nfan 1815 . . . . . . . . . 10 𝑦((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴)
449adantllr 750 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑦𝐵)
45 df-br 4578 . . . . . . . . . . . . . . . . 17 (𝑥𝑟𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑟)
46 eleq2 2676 . . . . . . . . . . . . . . . . . 18 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
47 opabid 4896 . . . . . . . . . . . . . . . . . 18 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
4846, 47syl6bb 274 . . . . . . . . . . . . . . . . 17 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
4945, 48syl5bb 270 . . . . . . . . . . . . . . . 16 (𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5049ad2antlr 758 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
51 elfvdm 6114 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑓𝑥) → 𝑥 ∈ dom 𝑓)
5251adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥 ∈ dom 𝑓)
53 fdm 5949 . . . . . . . . . . . . . . . . . . . . 21 (𝑓:𝐴⟶𝒫 𝐵 → dom 𝑓 = 𝐴)
545, 53syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → dom 𝑓 = 𝐴)
5554adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → dom 𝑓 = 𝐴)
5652, 55eleqtrd 2689 . . . . . . . . . . . . . . . . . 18 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝐴)
5756ex 448 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑦 ∈ (𝑓𝑥) → 𝑥𝐴))
5857pm4.71rd 664 . . . . . . . . . . . . . . . 16 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
5958ad2antrr 757 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
6050, 59bitr4d 269 . . . . . . . . . . . . . 14 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6160biimpar 500 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
6244, 61jca 552 . . . . . . . . . . . 12 ((((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → (𝑦𝐵𝑥𝑟𝑦))
6362ex 448 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) → (𝑦𝐵𝑥𝑟𝑦)))
6460biimpd 217 . . . . . . . . . . . 12 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑥𝑟𝑦𝑦 ∈ (𝑓𝑥)))
6564adantld 481 . . . . . . . . . . 11 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → ((𝑦𝐵𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥)))
6663, 65impbid 200 . . . . . . . . . 10 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
6743, 66abbid 2726 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = {𝑦 ∣ (𝑦𝐵𝑥𝑟𝑦)})
68 abid2 2731 . . . . . . . . . 10 {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥)
6968a1i 11 . . . . . . . . 9 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → {𝑦𝑦 ∈ (𝑓𝑥)} = (𝑓𝑥))
7037, 67, 693eqtr2rd 2650 . . . . . . . 8 (((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
7135, 70mpteq2da 4665 . . . . . . 7 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑥𝐴 ↦ (𝑓𝑥)) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7231, 71eqtrd 2643 . . . . . 6 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
7329, 72jca 552 . . . . 5 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
74 ssrab2 3649 . . . . . . . . . . . 12 {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵
7513elpw2 4749 . . . . . . . . . . . 12 ({𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵 ↔ {𝑦𝐵𝑥𝑟𝑦} ⊆ 𝐵)
7674, 75mpbir 219 . . . . . . . . . . 11 {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵
7776a1i 11 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝐴) → {𝑦𝐵𝑥𝑟𝑦} ∈ 𝒫 𝐵)
78 eqid 2609 . . . . . . . . . 10 (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}) = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
7977, 78fmptd 6276 . . . . . . . . 9 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
8079adantr 479 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)
81 simpr 475 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
8281feq1d 5928 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵))
8380, 82mpbird 245 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵)
8413pwex 4768 . . . . . . . 8 𝒫 𝐵 ∈ V
8584, 2elmap 7749 . . . . . . 7 (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)
8683, 85sylibr 222 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵𝑚 𝐴))
87 elpwi 4116 . . . . . . . . . 10 (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵))
8887adantr 479 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵))
89 xpss 5137 . . . . . . . . 9 (𝐴 × 𝐵) ⊆ (V × V)
9088, 89syl6ss 3579 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V))
91 df-rel 5034 . . . . . . . 8 (Rel 𝑟𝑟 ⊆ (V × V))
9290, 91sylibr 222 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel 𝑟)
93 relopab 5156 . . . . . . . 8 Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}
9493a1i 11 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
95 id 22 . . . . . . 7 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
96 nfv 1829 . . . . . . . . 9 𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
97 nfmpt1 4669 . . . . . . . . . 10 𝑥(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9897nfeq2 2765 . . . . . . . . 9 𝑥 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
9996, 98nfan 1815 . . . . . . . 8 𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
100 nfv 1829 . . . . . . . . 9 𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵)
10142nfci 2740 . . . . . . . . . . 11 𝑦𝐴
102 nfrab1 3098 . . . . . . . . . . 11 𝑦{𝑦𝐵𝑥𝑟𝑦}
103101, 102nfmpt 4668 . . . . . . . . . 10 𝑦(𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
104103nfeq2 2765 . . . . . . . . 9 𝑦 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})
105100, 104nfan 1815 . . . . . . . 8 𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))
106 nfcv 2750 . . . . . . . 8 𝑥𝑟
107 nfcv 2750 . . . . . . . 8 𝑦𝑟
108 brelg 28594 . . . . . . . . . . . . . . . 16 ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
10987, 108sylan 486 . . . . . . . . . . . . . . 15 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
110109adantlr 746 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦𝐵))
111110simpld 473 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝐴)
112110simprd 477 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦𝐵)
113 simpr 475 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦)
11481fveq1d 6089 . . . . . . . . . . . . . . . . . 18 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓𝑥) = ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥))
11513rabex 4734 . . . . . . . . . . . . . . . . . . 19 {𝑦𝐵𝑥𝑟𝑦} ∈ V
11678fvmpt2 6184 . . . . . . . . . . . . . . . . . . 19 ((𝑥𝐴 ∧ {𝑦𝐵𝑥𝑟𝑦} ∈ V) → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
117115, 116mpan2 702 . . . . . . . . . . . . . . . . . 18 (𝑥𝐴 → ((𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})‘𝑥) = {𝑦𝐵𝑥𝑟𝑦})
118114, 117sylan9eq 2663 . . . . . . . . . . . . . . . . 17 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑓𝑥) = {𝑦𝐵𝑥𝑟𝑦})
119118eleq2d 2672 . . . . . . . . . . . . . . . 16 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ 𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦}))
120 rabid 3094 . . . . . . . . . . . . . . . 16 (𝑦 ∈ {𝑦𝐵𝑥𝑟𝑦} ↔ (𝑦𝐵𝑥𝑟𝑦))
121119, 120syl6bb 274 . . . . . . . . . . . . . . 15 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
122111, 121syldan 485 . . . . . . . . . . . . . 14 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓𝑥) ↔ (𝑦𝐵𝑥𝑟𝑦)))
123112, 113, 122mpbir2and 958 . . . . . . . . . . . . 13 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓𝑥))
124111, 123jca 552 . . . . . . . . . . . 12 (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥𝐴𝑦 ∈ (𝑓𝑥)))
125124ex 448 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
126121simplbda 651 . . . . . . . . . . . 12 ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) ∧ 𝑥𝐴) ∧ 𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦)
127126expl 645 . . . . . . . . . . 11 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → ((𝑥𝐴𝑦 ∈ (𝑓𝑥)) → 𝑥𝑟𝑦))
128125, 127impbid 200 . . . . . . . . . 10 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
12945, 128syl5bbr 272 . . . . . . . . 9 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ (𝑥𝐴𝑦 ∈ (𝑓𝑥))))
130129, 47syl6bbr 276 . . . . . . . 8 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (⟨𝑥, 𝑦⟩ ∈ 𝑟 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13199, 105, 106, 107, 33, 39, 130eqrelrd2 28599 . . . . . . 7 (((Rel 𝑟 ∧ Rel {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13292, 94, 95, 131syl21anc 1316 . . . . . 6 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})
13386, 132jca 552 . . . . 5 ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}))
13473, 133impbii 197 . . . 4 ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦})))
135134a1i 11 . . 3 (⊤ → ((𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ∧ 𝑟 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥𝐴 ↦ {𝑦𝐵𝑥𝑟𝑦}))))
1361, 18, 20, 135f1od 6760 . 2 (⊤ → 𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵))
137136trud 1483 1 𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wtru 1475  wcel 1976  {cab 2595  {crab 2899  Vcvv 3172  wss 3539  𝒫 cpw 4107  cop 4130   class class class wbr 4577  {copab 4636  cmpt 4637   × cxp 5025  dom cdm 5027  Rel wrel 5032  wf 5785  1-1-ontowf1o 5788  cfv 5789  (class class class)co 6526  𝑚 cmap 7721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-1st 7036  df-2nd 7037  df-map 7723
This theorem is referenced by:  fpwrelmapffs  28690
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