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Theorem disjrdx 30971
Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)
Hypotheses
Ref Expression
disjrdx.1 (𝜑𝐹:𝐴1-1-onto𝐶)
disjrdx.2 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
Assertion
Ref Expression
disjrdx (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑦)

Proof of Theorem disjrdx
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 disjrdx.1 . . . . . . 7 (𝜑𝐹:𝐴1-1-onto𝐶)
2 f1of 6742 . . . . . . 7 (𝐹:𝐴1-1-onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelcdmda 6989 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 f1ofveu 7298 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐶𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
61, 5sylan 581 . . . . . 6 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
7 eqcom 2743 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87reubii 3337 . . . . . 6 (∃!𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
96, 8sylib 217 . . . . 5 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
10 disjrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
1110eleq2d 2822 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
124, 9, 11rmoxfrd 30882 . . . 4 (𝜑 → (∃*𝑦𝐶 𝑧𝐷 ↔ ∃*𝑥𝐴 𝑧𝐵))
1312bicomd 222 . . 3 (𝜑 → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐶 𝑧𝐷))
1413albidv 1921 . 2 (𝜑 → (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷))
15 df-disj 5047 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
16 df-disj 5047 . 2 (Disj 𝑦𝐶 𝐷 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷)
1714, 15, 163bitr4g 315 1 (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1537   = wceq 1539  wcel 2104  ∃!wreu 3282  ∃*wrmo 3283  Disj wdisj 5046  wf 6450  1-1-ontowf1o 6453  cfv 6454
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-disj 5047  df-br 5082  df-opab 5144  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462
This theorem is referenced by:  tocyccntz  31452  volmeas  32240  carsggect  32326
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