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Theorem ptunimpt 22204
 Description: Base set of a product topology given by substitution. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Hypothesis
Ref Expression
ptunimpt.j 𝐽 = (∏t‘(𝑥𝐴𝐾))
Assertion
Ref Expression
ptunimpt ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = 𝐽)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐽(𝑥)   𝐾(𝑥)   𝑉(𝑥)

Proof of Theorem ptunimpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqid 2801 . . . . . . . . 9 (𝑥𝐴𝐾) = (𝑥𝐴𝐾)
21fvmpt2 6760 . . . . . . . 8 ((𝑥𝐴𝐾 ∈ Top) → ((𝑥𝐴𝐾)‘𝑥) = 𝐾)
32eqcomd 2807 . . . . . . 7 ((𝑥𝐴𝐾 ∈ Top) → 𝐾 = ((𝑥𝐴𝐾)‘𝑥))
43unieqd 4817 . . . . . 6 ((𝑥𝐴𝐾 ∈ Top) → 𝐾 = ((𝑥𝐴𝐾)‘𝑥))
54ralimiaa 3130 . . . . 5 (∀𝑥𝐴 𝐾 ∈ Top → ∀𝑥𝐴 𝐾 = ((𝑥𝐴𝐾)‘𝑥))
65adantl 485 . . . 4 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → ∀𝑥𝐴 𝐾 = ((𝑥𝐴𝐾)‘𝑥))
7 ixpeq2 8462 . . . 4 (∀𝑥𝐴 𝐾 = ((𝑥𝐴𝐾)‘𝑥) → X𝑥𝐴 𝐾 = X𝑥𝐴 ((𝑥𝐴𝐾)‘𝑥))
86, 7syl 17 . . 3 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = X𝑥𝐴 ((𝑥𝐴𝐾)‘𝑥))
9 nffvmpt1 6660 . . . . 5 𝑥((𝑥𝐴𝐾)‘𝑦)
109nfuni 4810 . . . 4 𝑥 ((𝑥𝐴𝐾)‘𝑦)
11 nfcv 2958 . . . 4 𝑦 ((𝑥𝐴𝐾)‘𝑥)
12 fveq2 6649 . . . . 5 (𝑦 = 𝑥 → ((𝑥𝐴𝐾)‘𝑦) = ((𝑥𝐴𝐾)‘𝑥))
1312unieqd 4817 . . . 4 (𝑦 = 𝑥 ((𝑥𝐴𝐾)‘𝑦) = ((𝑥𝐴𝐾)‘𝑥))
1410, 11, 13cbvixp 8465 . . 3 X𝑦𝐴 ((𝑥𝐴𝐾)‘𝑦) = X𝑥𝐴 ((𝑥𝐴𝐾)‘𝑥)
158, 14eqtr4di 2854 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = X𝑦𝐴 ((𝑥𝐴𝐾)‘𝑦))
161fmpt 6855 . . 3 (∀𝑥𝐴 𝐾 ∈ Top ↔ (𝑥𝐴𝐾):𝐴⟶Top)
17 ptunimpt.j . . . 4 𝐽 = (∏t‘(𝑥𝐴𝐾))
1817ptuni 22203 . . 3 ((𝐴𝑉 ∧ (𝑥𝐴𝐾):𝐴⟶Top) → X𝑦𝐴 ((𝑥𝐴𝐾)‘𝑦) = 𝐽)
1916, 18sylan2b 596 . 2 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑦𝐴 ((𝑥𝐴𝐾)‘𝑦) = 𝐽)
2015, 19eqtrd 2836 1 ((𝐴𝑉 ∧ ∀𝑥𝐴 𝐾 ∈ Top) → X𝑥𝐴 𝐾 = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∪ cuni 4803   ↦ cmpt 5113  ⟶wf 6324  ‘cfv 6328  Xcixp 8448  ∏tcpt 16708  Topctop 21502 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-ixp 8449  df-en 8497  df-fin 8500  df-fi 8863  df-topgen 16713  df-pt 16714  df-top 21503  df-bases 21555 This theorem is referenced by:  pttopon  22205  kelac1  40000
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