f1eq1 - Metamath Proof Explorer

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Theorem f1eq1 6799

Description: Equality theorem for one-to-one functions. (Contributed by NM, 10-Feb-1997.)

**Assertion**
Ref	Expression
f1eq1	⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))

**Proof of Theorem f1eq1**
Step	Hyp	Ref	Expression
1		feq1 6716	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
2		cnveq 5884	. . . 4 ⊢ (𝐹 = 𝐺 → ^◡𝐹 = ^◡𝐺)
3	2	funeqd 6588	. . 3 ⊢ (𝐹 = 𝐺 → (Fun ^◡𝐹 ↔ Fun ^◡𝐺))
4	1, 3	anbi12d 632	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹) ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺)))
5		df-f1 6566	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
6		df-f1 6566	. 2 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ^◡ccnv 5684 Fun wfun 6555 ⟶wf 6557 –1-1→wf1 6558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566

This theorem is referenced by: f1oeq1 6836 f1eq123d 6840 fo00 6884 f1prex 7304 f1iun 7968 tposf12 8276 oacomf1olem 8602 f1dom4g 9006 f1dom3g 9008 f1domg 9012 dom3d 9034 domtr 9047 0domg 9140 domssex2 9177 1sdomOLD 9285 marypha1lem 9473 fseqenlem1 10064 dfac12lem2 10185 dfac12lem3 10186 ackbij2 10282 fin23lem28 10380 fin23lem32 10384 fin23lem34 10386 fin23lem35 10387 fin23lem41 10392 iundom2g 10580 pwfseqlem5 10703 hashf1lem1 14494 hashf1lem2 14495 hashf1 14496 4sqlem11 16993 injsubmefmnd 18910 conjsubgen 19269 sylow1lem2 19617 sylow2blem1 19638 hauspwpwf1 23995 istrkg2ld 28468 axlowdim 28976 sizusglecusg 29481 specval 31917 aciunf1lem 32672 zrhchr 33975 qqhre 34021 hashnexinj 42129 eldioph2lem2 42772 meadjiunlem 46480 fcoresf1b 47082 fundcmpsurbijinjpreimafv 47394 fundcmpsurinjpreimafv 47395 fundcmpsurinjimaid 47398 f1sn2g 48760 f102g 48761