Theorem foeq1 2784

Description: Equality theorem for onto functions.

Assertion

Ref	Expression
foeq1	⊢ (F = G → (F:A–onto→B ↔ G:A–onto→B))

Proof of Theorem foeq1

Step	Hyp	Ref	Expression
1		fneq1 2718	. . 3 ⊢ (F = G → (F Fn A ↔ G Fn A))
2		rneq 2555	. . . 4 ⊢ (F = G → ran F = ran G)
3	2	cleq1d 1109	. . 3 ⊢ (F = G → (ran F = B ↔ ran G = B))
4	1, 3	anbi12d 476	. 2 ⊢ (F = G → ((F Fn A ∧ ran F = B) ↔ (G Fn A ∧ ran G = B)))
5		df-fo 2436	. 2 ⊢ (F:A–onto→B ↔ (F Fn A ∧ ran F = B))
6		df-fo 2436	. 2 ⊢ (G:A–onto→B ↔ (G Fn A ∧ ran G = B))
7	4, 5, 6	3bitr4g 428	1 ⊢ (F = G → (F:A–onto→B ↔ G:A–onto→B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ran crn 2411   Fn wfn 2417  –onto→wfo 2420

This theorem is referenced by:  f1oeq1 2795  fo1st 3094  fo2nd 3095  fodomb 3615  ruclem39 4923  infmap2lem1 4951

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-fo 2436

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