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| Mirrors > Home > MPE Home > Th. List > mptmpoopabovdOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of mptmpoopabovd 8107 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 5154 | . 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ)) |
| 8 | fveq2 6906 | . . . 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 5154 | . 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 8108 | 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 1540 ∈ wcel 2108 Vcvv 3480 class class class wbr 5143 {copab 5205 ↦ cmpt 5225 ‘cfv 6561 (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 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| 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-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 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-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: wksonproplemOLD 29723 |
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