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Mirrors > Home > MPE Home > Th. List > nfimad | Structured version Visualization version GIF version |
Description: Deduction version of bound-variable hypothesis builder nfima 6058. (Contributed by FL, 15-Dec-2006.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfimad.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfimad.3 | ⊢ (𝜑 → Ⅎ𝑥𝐵) |
Ref | Expression |
---|---|
nfimad | ⊢ (𝜑 → Ⅎ𝑥(𝐴 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfaba1 2903 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} | |
2 | nfaba1 2903 | . . 3 ⊢ Ⅎ𝑥{𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} | |
3 | 1, 2 | nfima 6058 | . 2 ⊢ Ⅎ𝑥({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) |
4 | nfimad.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
5 | nfimad.3 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝐵) | |
6 | nfnfc1 2898 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐴 | |
7 | nfnfc1 2898 | . . . . 5 ⊢ Ⅎ𝑥Ⅎ𝑥𝐵 | |
8 | 6, 7 | nfan 1894 | . . . 4 ⊢ Ⅎ𝑥(Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) |
9 | abidnf 3691 | . . . . . 6 ⊢ (Ⅎ𝑥𝐴 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} = 𝐴) | |
10 | 9 | imaeq1d 6049 | . . . . 5 ⊢ (Ⅎ𝑥𝐴 → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵})) |
11 | abidnf 3691 | . . . . . 6 ⊢ (Ⅎ𝑥𝐵 → {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵} = 𝐵) | |
12 | 11 | imaeq2d 6050 | . . . . 5 ⊢ (Ⅎ𝑥𝐵 → (𝐴 “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) |
13 | 10, 12 | sylan9eq 2784 | . . . 4 ⊢ ((Ⅎ𝑥𝐴 ∧ Ⅎ𝑥𝐵) → ({𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐴} “ {𝑧 ∣ ∀𝑥 𝑧 ∈ 𝐵}) = (𝐴 “ 𝐵)) |
14 | 8, 13 | nfceqdf 2890 | . . 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)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 ∈ wcel 2098 {cab 2701 Ⅎwnfc 2875 “ cima 5670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-opab 5202 df-xp 5673 df-cnv 5675 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 |
This theorem is referenced by: (None) |
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