Theorem funfvdm2f 5259

Description: The value of a function. Version of funfvdm2 5258 using a bound-variable hypotheses instead of distinct variable conditions. (Contributed by Jim Kingdon, 1-Jan-2019.)

Hypotheses

Ref	Expression
funfvdm2f.1	⊢ Ⅎ𝑦𝐴
funfvdm2f.2	⊢ Ⅎ𝑦𝐹

Assertion

Ref	Expression
funfvdm2f	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Proof of Theorem funfvdm2f

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvdm2 5258	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑤 ∣ 𝐴𝐹𝑤})
2		funfvdm2f.1	. . . . 5 ⊢ Ⅎ𝑦𝐴
3		funfvdm2f.2	. . . . 5 ⊢ Ⅎ𝑦𝐹
4		nfcv 2219	. . . . 5 ⊢ Ⅎ𝑦𝑤
5	2, 3, 4	nfbr 3829	. . . 4 ⊢ Ⅎ𝑦 𝐴𝐹𝑤
6		nfv 1461	. . . 4 ⊢ Ⅎ𝑤 𝐴𝐹𝑦
7		breq2 3789	. . . 4 ⊢ (𝑤 = 𝑦 → (𝐴𝐹𝑤 ↔ 𝐴𝐹𝑦))
8	5, 6, 7	cbvab 2201	. . 3 ⊢ {𝑤 ∣ 𝐴𝐹𝑤} = {𝑦 ∣ 𝐴𝐹𝑦}
9	8	unieqi 3611	. 2 ⊢ ∪ {𝑤 ∣ 𝐴𝐹𝑤} = ∪ {𝑦 ∣ 𝐴𝐹𝑦}
10	1, 9	syl6eq 2129	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) = ∪ {𝑦 ∣ 𝐴𝐹𝑦})

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  {cab 2067  Ⅎwnfc 2206  ∪ cuni 3601   class class class wbr 3785  dom cdm 4363  Fun wfun 4916  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)