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Mirrors > Home > ILE Home > Th. List > foeq1 | GIF version |
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
foeq1 | ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1 5270 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴)) | |
2 | rneq 4825 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | eqeq1d 2173 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵)) |
4 | 1, 3 | anbi12d 465 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))) |
5 | df-fo 5188 | . 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵)) | |
6 | df-fo 5188 | . 2 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)) | |
7 | 4, 5, 6 | 3bitr4g 222 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1342 ran crn 4599 Fn wfn 5177 –onto→wfo 5180 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-fun 5184 df-fn 5185 df-fo 5188 |
This theorem is referenced by: f1oeq1 5415 foeq123d 5420 resdif 5448 dif1en 6836 0ct 7063 ctmlemr 7064 ctm 7065 ctssdclemn0 7066 ctssdclemr 7068 ctssdc 7069 enumct 7071 omct 7073 ctssexmid 7105 exmidfodomrlemim 7148 ennnfonelemim 12294 ctinfomlemom 12297 ctinfom 12298 ctinf 12300 qnnen 12301 enctlem 12302 ctiunct 12310 omctfn 12313 ssomct 12315 subctctexmid 13715 |
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