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Mirrors > Home > MPE Home > Th. List > prodex | Structured version Visualization version GIF version |
Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
Ref | Expression |
---|---|
prodex | ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-prod 15263 | . 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) | |
2 | iotaex 6338 | . 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) ∈ V | |
3 | 1, 2 | eqeltri 2912 | 1 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 ∃wrex 3142 Vcvv 3497 ⦋csb 3886 ⊆ wss 3939 ifcif 4470 class class class wbr 5069 ↦ cmpt 5149 ℩cio 6315 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 · cmul 10545 ℕcn 11641 ℤcz 11984 ℤ≥cuz 12246 ...cfz 12895 seqcseq 13372 ⇝ cli 14844 ∏cprod 15262 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-sn 4571 df-pr 4573 df-uni 4842 df-iota 6317 df-prod 15263 |
This theorem is referenced by: risefacval 15365 fallfacval 15366 prmoval 16372 fprodsubrecnncnvlem 42197 fprodaddrecnncnvlem 42199 etransclem13 42539 ovnlecvr 42847 ovncvrrp 42853 hoidmvval 42866 vonioolem1 42969 |
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