Theorem nfimad 4981

Description: Deduction version of bound-variable hypothesis builder nfima 4980. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
nfimad.2	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfimad.3	⊢ (𝜑 → Ⅎ𝑥𝐵)

Assertion

Ref	Expression
nfimad	⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))

Proof of Theorem nfimad

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfaba1 2325	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2325	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfima 4980	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})
4		nfimad.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfimad.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2322	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2322	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1565	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 2907	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	imaeq1d 4971	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}))
11		abidnf 2907	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	imaeq2d 4972	. . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
13	10, 12	sylan9eq 2230	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
14	8, 13	nfceqdf 2318	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
15	4, 5, 14	syl2anc 411	. 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
16	3, 15	mpbii 148	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   ∈ wcel 2148  {cab 2163  Ⅎwnfc 2306   “ cima 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641

This theorem is referenced by: (None)