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Mirrors > Home > MPE Home > Th. List > fvmptopab | Structured version Visualization version GIF version |
Description: The function value of a mapping 𝑀 to a restricted binary relation expressed as an ordered-pair class abstraction: The restricted binary relation is a binary relation given as value of a function 𝐹 restricted by the condition 𝜓. (Contributed by AV, 31-Jan-2021.) (Revised by AV, 29-Oct-2021.) Add disjoint variable condition on 𝐹, 𝑥, 𝑦 to remove a sethood hypothesis. (Revised by SN, 13-Dec-2024.) |
Ref | Expression |
---|---|
fvmptopab.1 | ⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓)) |
fvmptopab.m | ⊢ 𝑀 = (𝑧 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)}) |
Ref | Expression |
---|---|
fvmptopab | ⊢ (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6896 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍)) | |
2 | 1 | breqd 5160 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦)) |
3 | fvmptopab.1 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | anbi12d 630 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))) |
5 | 4 | opabbidv 5215 | . . 3 ⊢ (𝑧 = 𝑍 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)} = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
6 | fvmptopab.m | . . 3 ⊢ 𝑀 = (𝑧 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑)}) | |
7 | opabresex2 7472 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V | |
8 | 5, 6, 7 | fvmpt 7004 | . 2 ⊢ (𝑍 ∈ V → (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
9 | fvprc 6888 | . . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅) | |
10 | elopabran 5564 | . . . . . 6 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} → 𝑧 ∈ (𝐹‘𝑍)) | |
11 | 10 | ssriv 3980 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ (𝐹‘𝑍) |
12 | fvprc 6888 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅) | |
13 | 11, 12 | sseqtrid 4029 | . . . 4 ⊢ (¬ 𝑍 ∈ V → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅) |
14 | ss0 4400 | . . . 4 ⊢ ({〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅ → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (¬ 𝑍 ∈ V → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} = ∅) |
16 | 9, 15 | eqtr4d 2768 | . 2 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
17 | 8, 16 | pm2.61i 182 | 1 ⊢ (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⊆ wss 3944 ∅c0 4322 class class class wbr 5149 {copab 5211 ↦ cmpt 5232 ‘cfv 6549 |
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 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: trlsfval 29601 pthsfval 29627 spthsfval 29628 clwlks 29678 crcts 29694 cycls 29695 eupths 30102 |
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