Theorem funfv2f 5377

Description: The value of a function. Version of funfv2 5376 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 19-Feb-2006.)

Hypotheses

Ref	Expression
funfv2f.1	⊢ ℲyA
funfv2f.2	⊢ ℲyF

Assertion

Ref	Expression
funfv2f	⊢ (Fun F → (F ‘A) = ∪{y ∣ AFy})

Proof of Theorem funfv2f

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfv2 5376	. 2 ⊢ (Fun F → (F ‘A) = ∪{w ∣ AFw})
2		funfv2f.1	. . . . 5 ⊢ ℲyA
3		funfv2f.2	. . . . 5 ⊢ ℲyF
4		nfcv 2489	. . . . 5 ⊢ Ⅎyw
5	2, 3, 4	nfbr 4683	. . . 4 ⊢ Ⅎy AFw
6		nfv 1619	. . . 4 ⊢ Ⅎw AFy
7		breq2 4643	. . . 4 ⊢ (w = y → (AFw ↔ AFy))
8	5, 6, 7	cbvab 2471	. . 3 ⊢ {w ∣ AFw} = {y ∣ AFy}
9	8	unieqi 3901	. 2 ⊢ ∪{w ∣ AFw} = ∪{y ∣ AFy}
10	1, 9	syl6eq 2401	1 ⊢ (Fun F → (F ‘A) = ∪{y ∣ AFy})

Syntax hints:   → wi 4   = wceq 1642  {cab 2339  Ⅎwnfc 2476  ∪cuni 3891   class class class wbr 4639  Fun wfun 4775   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by: (None)