Theorem funfv2f 6746

Description: The value of a function. Version of funfv2 6745 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 6745	. 2 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfv2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfv2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2977	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 5105	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1911	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 5062	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvabw 2890	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 4840	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	syl6eq 2872	1 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   = wceq 1533  {cab 2799  Ⅎwnfc 2961  ∪ cuni 4831   class class class wbr 5058  Fun wfun 6343  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by: (None)