![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mptmpoopabovdOLD | Structured version Visualization version GIF version |
Description: Obsolete version of mptmpoopabovd 8013 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 7365 | . . 3 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑎(𝐶‘𝐺)𝑏) = (𝑋(𝐶‘𝐺)𝑌)) | |
7 | 6 | breqd 5116 | . 2 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑓(𝑎(𝐶‘𝐺)𝑏)ℎ ↔ 𝑓(𝑋(𝐶‘𝐺)𝑌)ℎ)) |
8 | fveq2 6842 | . . . 4 ⊢ (𝑔 = 𝐺 → (𝐶‘𝑔) = (𝐶‘𝐺)) | |
9 | 8 | oveqd 7373 | . . 3 ⊢ (𝑔 = 𝐺 → (𝑎(𝐶‘𝑔)𝑏) = (𝑎(𝐶‘𝐺)𝑏)) |
10 | 9 | breqd 5116 | . 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 | mptmpoopabbrdOLD 8014 | 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 class class class wbr 5105 {copab 5167 ↦ cmpt 5188 ‘cfv 6496 (class class class)co 7356 ∈ cmpo 7358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 |
This theorem is referenced by: wksonproplemOLD 28600 |
Copyright terms: Public domain | W3C validator |