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Mirrors > Home > MPE Home > Th. List > fvmptd3f | Structured version Visualization version GIF version |
Description: Alternate deduction version of fvmpt 6768 with three non-freeness 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 5164 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐷 ↦ 𝐵) | |
4 | 2, 3 | nfeq 2991 | . . 3 ⊢ Ⅎ𝑥 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) |
5 | fvmptd3f.5 | . . 3 ⊢ Ⅎ𝑥𝜓 | |
6 | 4, 5 | nfim 1897 | . 2 ⊢ Ⅎ𝑥(𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓) |
7 | fvmptd2f.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
8 | 7 | elexd 3514 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) |
9 | isset 3506 | . . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 8, 9 | sylib 220 | . 2 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴) |
11 | fveq1 6669 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) | |
12 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴) | |
13 | 12 | fveq2d 6674 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
14 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐴 ∈ 𝐷) |
15 | 12, 14 | eqeltrd 2913 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 ∈ 𝐷) |
16 | fvmptd2f.2 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝑉) | |
17 | eqid 2821 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
18 | 17 | fvmpt2 6779 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐷 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
19 | 15, 16, 18 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝑥) = 𝐵) |
20 | 13, 19 | eqtr3d 2858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = 𝐵) |
21 | 20 | eqeq2d 2832 | . . . 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 242 | . . 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 2220 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵) → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 Vcvv 3494 ↦ cmpt 5146 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 |
This theorem is referenced by: fvmptd2f 6784 |
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