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Mirrors > Home > MPE Home > Th. List > mptmpoopabovdOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mptmpoopabovd 8106 as of 13-Dec-2024. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 15-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mptmpoopabbrdOLD.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
mptmpoopabbrdOLD.x | ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) |
mptmpoopabbrdOLD.y | ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) |
mptmpoopabbrdOLD.v | ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) |
mptmpoopabbrdOLD.r | ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) |
mptmpoopabovdOLD.m | ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) |
Ref | Expression |
---|---|
mptmpoopabovdOLD | ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptmpoopabbrdOLD.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
2 | mptmpoopabbrdOLD.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝐴‘𝐺)) | |
3 | mptmpoopabbrdOLD.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝐵‘𝐺)) | |
4 | mptmpoopabbrdOLD.v | . 2 ⊢ (𝜑 → {〈𝑓, ℎ〉 ∣ 𝜓} ∈ 𝑉) | |
5 | mptmpoopabbrdOLD.r | . 2 ⊢ ((𝜑 ∧ 𝑓(𝐷‘𝐺)ℎ) → 𝜓) | |
6 | oveq12 7440 | . . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌)) | |
7 | 6 | breqd 5159 | . 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ)) |
8 | fveq2 6907 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺)) | |
9 | 8 | oveqd 7448 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏)) |
10 | 9 | breqd 5159 | . 2 ⊢ (𝑔 = 𝐺 → (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ↔ 𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ)) |
11 | mptmpoopabovdOLD.m | . 2 ⊢ 𝑀 = (𝑔 ∈ V ↦ (𝑎 ∈ (𝐴‘𝑔), 𝑏 ∈ (𝐵‘𝑔) ↦ {〈𝑓, ℎ〉 ∣ (𝑓(𝑎(𝐶‘𝑔)𝑏)ℎ ∧ 𝑓(𝐷‘𝑔)ℎ)})) | |
12 | 1, 2, 3, 4, 5, 7, 10, 11 | mptmpoopabbrdOLDOLD 8107 | 1 ⊢ (𝜑 → (𝑋(𝑀‘𝐺)𝑌) = {〈𝑓, ℎ〉 ∣ (𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ ∧ 𝑓(𝐷‘𝐺)ℎ)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 class class class wbr 5148 {copab 5210 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 |
This theorem is referenced by: wksonproplemOLD 29738 |
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