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Mirrors > Home > MPE Home > Th. List > fvmptopab1 | 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.) |
Ref | Expression |
---|---|
fvmptopab1.2 | ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓)) |
fvmptopab1.3 | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
fvmptopab1.4 | ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V) |
fvmptopab1.1 | ⊢ 𝑀 = (𝑧 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)}) |
Ref | Expression |
---|---|
fvmptopab1 | ⊢ (𝜑 → (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptopab1.1 | . . 3 ⊢ 𝑀 = (𝑧 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)}) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → 𝑀 = (𝑧 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)})) |
3 | fveq2 6103 | . . . . . 6 ⊢ (𝑧 = 𝑍 → (𝐹‘𝑧) = (𝐹‘𝑍)) | |
4 | 3 | breqd 4594 | . . . . 5 ⊢ (𝑧 = 𝑍 → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦)) |
5 | 4 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝑥(𝐹‘𝑧)𝑦 ↔ 𝑥(𝐹‘𝑍)𝑦)) |
6 | fvmptopab1.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → (𝜒 ↔ 𝜓)) | |
7 | 5, 6 | anbi12d 743 | . . 3 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → ((𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒) ↔ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓))) |
8 | 7 | opabbidv 4648 | . 2 ⊢ ((𝜑 ∧ 𝑧 = 𝑍) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑧)𝑦 ∧ 𝜒)} = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
9 | fvmptopab1.3 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
10 | 9 | elexd 3187 | . 2 ⊢ (𝜑 → 𝑍 ∈ V) |
11 | id 22 | . . . 4 ⊢ (𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦) | |
12 | 11 | gen2 1714 | . . 3 ⊢ ∀𝑥∀𝑦(𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦) |
13 | fvmptopab1.4 | . . 3 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V) | |
14 | opabbrex 6593 | . . 3 ⊢ ((∀𝑥∀𝑦(𝑥(𝐹‘𝑍)𝑦 → 𝑥(𝐹‘𝑍)𝑦) ∧ {〈𝑥, 𝑦〉 ∣ 𝑥(𝐹‘𝑍)𝑦} ∈ V) → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V) | |
15 | 12, 13, 14 | sylancr 694 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)} ∈ V) |
16 | 2, 8, 10, 15 | fvmptd 6197 | 1 ⊢ (𝜑 → (𝑀‘𝑍) = {〈𝑥, 𝑦〉 ∣ (𝑥(𝐹‘𝑍)𝑦 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: fvmptopab2 6595 trlsfval 40903 spthsfval 40928 clwlkS 40978 crctS 40994 cyclS 40995 eupths 41367 |
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