Theorem funeq 6501

Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
funeq	⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Proof of Theorem funeq

Step	Hyp	Ref	Expression
1		eqimss2 3989	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6500	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3988	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6500	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 17	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 212	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ⊆ wss 3897  Fun wfun 6475

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ss 3914  df-br 5090  df-opab 5152  df-rel 5621  df-cnv 5622  df-co 5623  df-fun 6483

This theorem is referenced by:  funeqi  6502  funeqd  6503  fununi  6556  cnvresid  6560  fneq1  6572  funop  7082  funsndifnop  7084  nvof1o  7214  funcnvuni  7862  fiun  7875  elpmg  8767  fundmeng  8954  isfsupp  9249  dfac9  10028  axdc3lem2  10342  frlmphllem  21717  psdmul  22081  oldval  27795  usgredgop  29148  locfinreflem  33853  orvcval  34471  bnj1379  34842  bnj1385  34844  bnj1497  35072  funen1cnv  35100  elfunsg  35958  modelaxreplem1  45081  modelaxreplem2  45082  modelaxrep  45084  funop1  47393