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Mirrors > Home > MPE Home > Th. List > fvmptd3f | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 7015 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.) |
Ref | Expression |
---|---|
fvmptd2f.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd2f.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptd2f.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) |
fvmptd3f.4 | ⊢ Ⅎ𝑥𝐹 |
fvmptd3f.5 | ⊢ Ⅎ𝑥𝜓 |
fvmptd3f.6 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
fvmptd3f | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd3f.6 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | fvmptd3f.4 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nfmpt1 5255 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
4 | 2, 3 | nfeq 2916 | . . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
5 | fvmptd3f.5 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfim 1893 | . 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓) |
7 | fvmptd2f.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
8 | 7 | elexd 3501 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
9 | isset 3491 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 8, 9 | sylib 218 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
11 | fveq1 6905 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
12 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
13 | 12 | fveq2d 6910 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
14 | 7 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐷) |
15 | 12, 14 | eqeltrd 2838 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷) |
16 | fvmptd2f.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
17 | eqid 2734 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
18 | 17 | fvmpt2 7026 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
19 | 15, 16, 18 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
20 | 13, 19 | eqtr3d 2776 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵) |
21 | 20 | eqeq2d 2745 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = 𝐵)) |
22 | fvmptd2f.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) | |
23 | 21, 22 | sylbid 240 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) → 𝜓)) |
24 | 11, 23 | syl5 34 | . 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
25 | 1, 6, 10, 24 | exlimdd 2217 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∃wex 1775 Ⅎwnf 1779 ∈ wcel 2105 Ⅎwnfc 2887 Vcvv 3477 ↦ cmpt 5230 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fv 6570 |
This theorem is referenced by: fvmptd2f 7031 |
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