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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 6906 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍)) | |
| 2 | 1 | breqd 5154 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦)) |
| 3 | fvmptopab.1 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝜑 ↔ 𝜓)) | |
| 4 | 2, 3 | anbi12d 632 | . . . 4 ⊢ (𝑧 = 𝑍 → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜑) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))) |
| 5 | 4 | opabbidv 5209 | . . 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 7485 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V | |
| 8 | 5, 6, 7 | fvmpt 7016 | . 2 ⊢ (𝑍 ∈ V → (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
| 9 | fvprc 6898 | . . 3 ⊢ (¬ 𝑍 ∈ V → (𝑀‘𝑍) = ∅) | |
| 10 | elopabran 5567 | . . . . . 6 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} → 𝑧 ∈ (𝐹‘𝑍)) | |
| 11 | 10 | ssriv 3987 | . . . . 5 ⊢ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ (𝐹‘𝑍) |
| 12 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → (𝐹‘𝑍) = ∅) | |
| 13 | 11, 12 | sseqtrid 4026 | . . . 4 ⊢ (¬ 𝑍 ∈ V → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ⊆ ∅) |
| 14 | ss0 4402 | . . . 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 2780 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ‘cfv 6561 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 |
| This theorem is referenced by: trlsfval 29713 pthsfval 29739 spthsfval 29740 clwlks 29792 crcts 29808 cycls 29809 eupths 30219 |
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