Theorem funfv2f 7011

Description: The value of a function. Version of funfv2 7010 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 7010	. 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfv2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfv2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 5213	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1913	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 5170	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvabw 2816	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 4943	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	eqtrdi 2796	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   = wceq 1537  {cab 2717  Ⅎwnfc 2893  ∪ cuni 4931   class class class wbr 5166  Fun wfun 6567  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by: (None)