![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6885 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
2 | fvmptrabfv.r | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | rabeqbidv 3443 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
4 | fvmptrabfv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) | |
5 | fvex 6898 | . . . 4 ⊢ (𝐺‘𝑋) ∈ V | |
6 | 5 | rabex 5325 | . . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V |
7 | 3, 4, 6 | fvmpt 6992 | . 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
8 | fvprc 6877 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | fvprc 6877 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅) | |
10 | 9 | rabeqdv 3441 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) |
11 | rab0 4377 | . . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
12 | 10, 11 | eqtr2di 2783 | . . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
13 | 8, 12 | eqtrd 2766 | . 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 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ∅c0 4317 ↦ cmpt 5224 ‘cfv 6537 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: uvtxval 29152 |
Copyright terms: Public domain | W3C validator |