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Theorem disjrdx 29530
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 6175 . . . . . . 7 (𝐹:𝐴1-1-onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelrnda 6399 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 f1ofveu 6685 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐶𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
61, 5sylan 487 . . . . . 6 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
7 eqcom 2658 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87reubii 3158 . . . . . 6 (∃!𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
96, 8sylib 208 . . . . 5 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
10 disjrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
1110eleq2d 2716 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
124, 9, 11rmoxfrd 29460 . . . 4 (𝜑 → (∃*𝑦𝐶 𝑧𝐷 ↔ ∃*𝑥𝐴 𝑧𝐵))
1312bicomd 213 . . 3 (𝜑 → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐶 𝑧𝐷))
1413albidv 1889 . 2 (𝜑 → (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷))
15 df-disj 4653 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
16 df-disj 4653 . 2 (Disj 𝑦𝐶 𝐷 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷)
1714, 15, 163bitr4g 303 1 (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  ∃!wreu 2943  ∃*wrmo 2944  Disj wdisj 4652  wf 5922  1-1-ontowf1o 5925  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-disj 4653  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934
This theorem is referenced by:  volmeas  30422  carsggect  30508
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