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Mirrors > Home > MPE Home > Th. List > fvmptd3f | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6988 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 5246 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
4 | 2, 3 | nfeq 2908 | . . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
5 | fvmptd3f.5 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfim 1891 | . 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓) |
7 | fvmptd2f.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
8 | 7 | elexd 3487 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
9 | isset 3479 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
11 | fveq1 6880 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
12 | simpr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
13 | 12 | fveq2d 6885 | . . . . . 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 2825 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷) |
16 | fvmptd2f.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
17 | eqid 2724 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
18 | 17 | fvmpt2 6999 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
19 | 15, 16, 18 | syl2anc 583 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
20 | 13, 19 | eqtr3d 2766 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵) |
21 | 20 | eqeq2d 2735 | . . . 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 239 | . . 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 2205 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2875 Vcvv 3466 ↦ cmpt 5221 ‘cfv 6533 |
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 ax-sep 5289 ax-nul 5296 ax-pr 5417 |
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-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fv 6541 |
This theorem is referenced by: fvmptd2f 7004 |
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