Theorem nfimad 6012

Description: Deduction version of bound-variable hypothesis builder nfima 6011. (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 2913	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2913	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfima 6011	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})
4		nfimad.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfimad.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2908	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2908	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1902	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3652	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	imaeq1d 6002	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}))
11		abidnf 3652	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	imaeq2d 6003	. . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
13	10, 12	sylan9eq 2797	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
14	8, 13	nfceqdf 2900	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
15	4, 5, 14	syl2anc 585	. 2 ⊢ (𝜑 → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
16	3, 15	mpbii 232	1 ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1539   ∈ wcel 2106  {cab 2714  Ⅎwnfc 2885   “ cima 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-br 5097  df-opab 5159  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by: (None)