Theorem feq1 2748

Description: Equality theorem for functions.

Assertion

Ref	Expression
feq1	⊢ (F = G → (F:A–→B ↔ G:A–→B))

Proof of Theorem feq1

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	sseq1d 1527	. . 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-f 2434	. 2 ⊢ (F:A–→B ↔ (F Fn A ∧ ran F ⊆ B))
6		df-f 2434	. 2 ⊢ (G:A–→B ↔ (G Fn A ∧ ran G ⊆ B))
7	4, 5, 6	3bitr4g 428	1 ⊢ (F = G → (F:A–→B ↔ G:A–→B))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ⊆ wss 1487  ran crn 2411   Fn wfn 2417  –→wf 2418

This theorem is referenced by:  f00 2773  fconst 2774  fconstg 2775  f1eq1 2776  fopab2 2891  fressnfv 2898  fconst2 2902  fconstfv 2903  elmap 3265  map0 3268  pw2en 3348  xpmapenlem4 3394  ac6lem 3575  clim 4877  clim2 4881  ruclem13 4897  ruclem17 4901  ruclem39 4923  hcauchy 5103  hlim 5108  hlim2 5112  chlim 5139  chcompl 5150  hosmvalt 5487  hodmvalt 5488  stelt 5671

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-f 2434

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