Theorem funeq 6587

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 3972	. . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴)
2		funss 6586	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (Fun 𝐴 → Fun 𝐵))
3	1, 2	syl 18	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 → Fun 𝐵))
4		eqimss 3971	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵)
5		funss 6586	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (Fun 𝐵 → Fun 𝐴))
6	4, 5	syl 18	. 2 ⊢ (𝐴 = 𝐵 → (Fun 𝐵 → Fun 𝐴))
7	3, 6	impbid 215	1 ⊢ (𝐴 = 𝐵 → (Fun 𝐴 ↔ Fun 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ⊆ wss 3900  Fun wfun 6560

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-fun 6568

This theorem is used by:  funeqi  6588  funeqd  6589  fununi  6645  cnvresidOLD  6649  fneq1  6660  funop  7185  funsndifnop  7187  nvof1o  7319  funcnvuni  7978  fiun  7989  elpmg  8901  funen1cnv  9094  fundmeng  9098  isfsupp  9395  dfac9  10254  axdc3lem2  10568  frlmphllem  22141  psdmul  22542  oldval  28277  usgredgop  29808  locfinreflem  34583  orvcval  35201  bnj1379  35571  bnj1385  35573  bnj1497  35801  srepdflem2  36127  srepdf  36128  elfunsg  36893  modelaxreplem1  46232  modelaxreplem2  46233  modelaxrep  46235  funop1  48626