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Mirrors > Home > MPE Home > Th. List > fvmptdf | Structured version Visualization version GIF version |
Description: Deduction version of fvmptd 6752 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by AV, 29-Mar-2024.) |
Ref | Expression |
---|---|
fvmptd.1 | ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) |
fvmptd.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) |
fvmptd.3 | ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
fvmptd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
fvmptdf.p | ⊢ Ⅎ𝑥𝜑 |
fvmptdf.a | ⊢ Ⅎ𝑥𝐴 |
fvmptdf.c | ⊢ Ⅎ𝑥𝐶 |
Ref | Expression |
---|---|
fvmptdf | ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptd.1 | . . 3 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐷 ↦ 𝐵)) | |
2 | 1 | fveq1d 6647 | . 2 ⊢ (𝜑 → (𝐹‘𝐴) = ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴)) |
3 | fvmptd.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐷) | |
4 | fvmptdf.p | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
5 | nfcsb1v 3852 | . . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 | |
6 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵) |
7 | fvmptdf.c | . . . . . 6 ⊢ Ⅎ𝑥𝐶 | |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝐶) |
9 | csbeq1a 3842 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) | |
10 | 9 | adantl 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵) |
11 | fvmptdf.a | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
12 | 11 | nfeq2 2972 | . . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = 𝐴 |
13 | 4, 12 | nfan 1900 | . . . . . 6 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑦 = 𝐴) |
14 | 7 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → Ⅎ𝑥𝐶) |
15 | vex 3444 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 ∈ V) |
17 | eqtr 2818 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑦 = 𝐴) → 𝑥 = 𝐴) | |
18 | 17 | ancoms 462 | . . . . . . . 8 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝑦) → 𝑥 = 𝐴) |
19 | fvmptd.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 = 𝐶) | |
20 | 18, 19 | sylan2 595 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝑦)) → 𝐵 = 𝐶) |
21 | 20 | anassrs 471 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 = 𝐴) ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) |
22 | 13, 14, 16, 21 | csbiedf 3858 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 = 𝐶) |
23 | 4, 6, 8, 3, 10, 22 | csbie2df 4348 | . . . 4 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 = 𝐶) |
24 | fvmptd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
25 | 23, 24 | eqeltrd 2890 | . . 3 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) |
26 | eqid 2798 | . . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ 𝐵) = (𝑥 ∈ 𝐷 ↦ 𝐵) | |
27 | 26 | fvmpts 6748 | . . 3 ⊢ ((𝐴 ∈ 𝐷 ∧ ⦋𝐴 / 𝑥⦌𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
28 | 3, 25, 27 | syl2anc 587 | . 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐷 ↦ 𝐵)‘𝐴) = ⦋𝐴 / 𝑥⦌𝐵) |
29 | 2, 28, 23 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (𝐹‘𝐴) = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 Ⅎwnfc 2936 Vcvv 3441 ⦋csb 3828 ↦ cmpt 5110 ‘cfv 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 |
This theorem is referenced by: fvmptd 6752 symgval 18489 1arymaptfo 45057 2arymaptfo 45068 |
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