Theorem nfimad 5967

Description: Deduction version of bound-variable hypothesis builder nfima 5966. (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 2914	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴}
2		nfaba1 2914	. . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}
3	1, 2	nfima 5966	. 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})
4		nfimad.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴)
5		nfimad.3	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵)
6		nfnfc1 2909	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴
7		nfnfc1 2909	. . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵
8	6, 7	nfan 1903	. . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵)
9		abidnf 3633	. . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴)
10	9	imaeq1d 5957	. . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}))
11		abidnf 3633	. . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵)
12	11	imaeq2d 5958	. . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
13	10, 12	sylan9eq 2799	. . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵))
14	8, 13	nfceqdf 2901	. . 3 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → (Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) ↔ Ⅎ𝑥(𝐴 “ 𝐵)))
15	4, 5, 14	syl2anc 583	. 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 395  ∀wal 1537   ∈ wcel 2108  {cab 2715  Ⅎwnfc 2886   “ cima 5583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593

This theorem is referenced by: (None)