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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 6538 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋)) | |
2 | fvmptrabfv.r | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | rabeqbidv 3430 | . . 3 ⊢ (𝑥 = 𝑋 → {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑} = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
4 | fvmptrabfv.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {𝑦 ∈ (𝐺‘𝑥) ∣ 𝜑}) | |
5 | fvex 6551 | . . . 4 ⊢ (𝐺‘𝑋) ∈ V | |
6 | 5 | rabex 5126 | . . 3 ⊢ {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} ∈ V |
7 | 3, 4, 6 | fvmpt 6635 | . 2 ⊢ (𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
8 | fvprc 6531 | . . 3 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = ∅) | |
9 | fvprc 6531 | . . . . 5 ⊢ (¬ 𝑋 ∈ V → (𝐺‘𝑋) = ∅) | |
10 | 9 | rabeqdv 3429 | . . . 4 ⊢ (¬ 𝑋 ∈ V → {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} = {𝑦 ∈ ∅ ∣ 𝜓}) |
11 | rab0 4257 | . . . 4 ⊢ {𝑦 ∈ ∅ ∣ 𝜓} = ∅ | |
12 | 10, 11 | syl6req 2848 | . . 3 ⊢ (¬ 𝑋 ∈ V → ∅ = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
13 | 8, 12 | eqtrd 2831 | . 2 ⊢ (¬ 𝑋 ∈ V → (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓}) |
14 | 7, 13 | pm2.61i 183 | 1 ⊢ (𝐹‘𝑋) = {𝑦 ∈ (𝐺‘𝑋) ∣ 𝜓} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 = wceq 1522 ∈ wcel 2081 {crab 3109 Vcvv 3437 ∅c0 4211 ↦ cmpt 5041 ‘cfv 6225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-iota 6189 df-fun 6227 df-fv 6233 |
This theorem is referenced by: uvtxval 26852 |
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