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Theorem cnvf1olem 6129

Description: Lemma for cnvf1o 6130. (Contributed by Mario Carneiro, 27-Apr-2014.)

**Assertion**
Ref	Expression
cnvf1olem	⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

**Proof of Theorem cnvf1olem**
Step	Hyp	Ref	Expression
1		simprr 522	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ∪ ^◡{𝐵})
2		1st2nd 6087	. . . . . . . 8 ⊢ ((Rel 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
3	2	adantrr 471	. . . . . . 7 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
4	3	sneqd 3545	. . . . . 6 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐵} = {⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
5	4	cnveqd 4723	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐵} = ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
6	5	unieqd 3755	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐵} = ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩})
7		1stexg 6073	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (1^st ‘𝐵) ∈ V)
8		2ndexg 6074	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (2^nd ‘𝐵) ∈ V)
9		opswapg 5033	. . . . . 6 ⊢ (((1^st ‘𝐵) ∈ V ∧ (2^nd ‘𝐵) ∈ V) → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
10	7, 8, 9	syl2anc 409	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
11	10	ad2antrl 482	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩} = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
12	1, 6, 11	3eqtrd 2177	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 = ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩)
13		simprl 521	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 ∈ 𝐴)
14	3, 13	eqeltrrd 2218	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴)
15		opelcnvg 4727	. . . . . 6 ⊢ (((2^nd ‘𝐵) ∈ V ∧ (1^st ‘𝐵) ∈ V) → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
16	8, 7, 15	syl2anc 409	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
17	16	ad2antrl 482	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴 ↔ ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ 𝐴))
18	14, 17	mpbird 166	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩ ∈ ^◡𝐴)
19	12, 18	eqeltrd 2217	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐶 ∈ ^◡𝐴)
20		opswapg 5033	. . . . . 6 ⊢ (((2^nd ‘𝐵) ∈ V ∧ (1^st ‘𝐵) ∈ V) → ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
21	8, 7, 20	syl2anc 409	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩} = ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩)
22	21	eqcomd 2146	. . . 4 ⊢ (𝐵 ∈ 𝐴 → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
23	22	ad2antrl 482	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
24	12	sneqd 3545	. . . . 5 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → {𝐶} = {⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
25	24	cnveqd 4723	. . . 4 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ^◡{𝐶} = ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
26	25	unieqd 3755	. . 3 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → ∪ ^◡{𝐶} = ∪ ^◡{⟨(2^nd ‘𝐵), (1^st ‘𝐵)⟩})
27	23, 3, 26	3eqtr4d 2183	. 2 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → 𝐵 = ∪ ^◡{𝐶})
28	19, 27	jca 304	1 ⊢ ((Rel 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 = ∪ ^◡{𝐵})) → (𝐶 ∈ ^◡𝐴 ∧ 𝐵 = ∪ ^◡{𝐶}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 ⟨cop 3535 ∪ cuni 3744 ^◡ccnv 4546 Rel wrel 4552 ‘cfv 5131 1^st c1st 6044 2^nd c2nd 6045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-fo 5137 df-fv 5139 df-1st 6046 df-2nd 6047

This theorem is referenced by: cnvf1o 6130

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