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Mirrors > Home > MPE Home > Th. List > fvmptrabfv | Structured version Visualization version GIF version |
Description: Value of a function mapping a set to a class abstraction restricting the value of another function. (Contributed by AV, 18-Feb-2022.) |
Ref | Expression |
---|---|
fvmptrabfv.f | ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) |
fvmptrabfv.r | ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
fvmptrabfv | ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6881 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
2 | fvmptrabfv.r | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | rabeqbidv 3450 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
4 | fvmptrabfv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) | |
5 | fvex 6894 | . . . 4 ⊢ (𝐺‘𝑋) ∈ V | |
6 | 5 | rabex 5328 | . . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V |
7 | 3, 4, 6 | fvmpt 6987 | . 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
8 | fvprc 6873 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | fvprc 6873 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅) | |
10 | 9 | rabeqdv 3448 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) |
11 | rab0 4380 | . . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
12 | 10, 11 | eqtr2di 2790 | . . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
13 | 8, 12 | eqtrd 2773 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
14 | 7, 13 | pm2.61i 182 | 1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ∅c0 4320 ↦ cmpt 5227 ‘cfv 6535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6487 df-fun 6537 df-fv 6543 |
This theorem is referenced by: uvtxval 28611 |
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