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Mirrors > Home > MPE Home > Th. List > fvmptd3f | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6597 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.) |
Ref | Expression |
---|---|
fvmptdf.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptdf.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) |
fvmptdf.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 5026 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
4 | 2, 3 | nfeq 2943 | . . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
5 | fvmptd3f.5 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfim 1859 | . 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓) |
7 | fvmptdf.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
8 | 7 | elexd 3435 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
9 | isset 3427 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 8, 9 | sylib 210 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
11 | fveq1 6500 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
12 | simpr 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
13 | 12 | fveq2d 6505 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
14 | 7 | adantr 473 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐷) |
15 | 12, 14 | eqeltrd 2866 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷) |
16 | fvmptdf.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
17 | eqid 2778 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
18 | 17 | fvmpt2 6607 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
19 | 15, 16, 18 | syl2anc 576 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
20 | 13, 19 | eqtr3d 2816 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵) |
21 | 20 | eqeq2d 2788 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) ↔ (𝐹‘𝐴) = 𝐵)) |
22 | fvmptdf.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝐹‘𝐴) = 𝐵 → 𝜓)) | |
23 | 21, 22 | sylbid 232 | . . 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 2150 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∃wex 1742 Ⅎwnf 1746 ∈ wcel 2050 Ⅎwnfc 2916 Vcvv 3415 ↦ cmpt 5009 ‘cfv 6190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fv 6198 |
This theorem is referenced by: fvmptdf 6612 |
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