f1eq1 - Metamath Proof Explorer

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Theorem f1eq1 6572

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 6497	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶𝐵 ↔ 𝐺:𝐴⟶𝐵))
2		cnveq 5746	. . . 4 ⊢ (𝐹 = 𝐺 → ^◡𝐹 = ^◡𝐺)
3	2	funeqd 6379	. . 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 6362	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ Fun ^◡𝐹))
6		df-f1 6362	. 2 ⊢ (𝐺:𝐴–1-1→𝐵 ↔ (𝐺:𝐴⟶𝐵 ∧ Fun ^◡𝐺))
7	4, 5, 6	3bitr4g 316	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1→𝐵 ↔ 𝐺:𝐴–1-1→𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ^◡ccnv 5556 Fun wfun 6351 ⟶wf 6353 –1-1→wf1 6354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362

This theorem is referenced by: f1oeq1 6606 f1eq123d 6610 fo00 6652 f1prex 7042 f1iun 7647 tposf12 7919 oacomf1olem 8192 f1dom2g 8529 f1domg 8531 dom3d 8553 domtr 8564 domssex2 8679 1sdom 8723 marypha1lem 8899 fseqenlem1 9452 dfac12lem2 9572 dfac12lem3 9573 ackbij2 9667 fin23lem28 9764 fin23lem32 9768 fin23lem34 9770 fin23lem35 9771 fin23lem41 9776 iundom2g 9964 pwfseqlem5 10087 hashf1lem1 13816 hashf1lem2 13817 hashf1 13818 4sqlem11 16293 injsubmefmnd 18064 conjsubgen 18393 sylow1lem2 18726 sylow2blem1 18747 hauspwpwf1 22597 istrkg2ld 26248 axlowdim 26749 sizusglecusg 27247 specval 29677 aciunf1lem 30409 zrhchr 31219 qqhre 31263 eldioph2lem2 39365 meadjiunlem 42754 fundcmpsurbijinjpreimafv 43574 fundcmpsurinjpreimafv 43575 fundcmpsurinjimaid 43578