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Theorem disjrdx 30333
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 6608 . . . . . . 7 (𝐹:𝐴1-1-onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelrnda 6844 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 f1ofveu 7143 . . . . . . 7 ((𝐹:𝐴1-1-onto𝐶𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
61, 5sylan 582 . . . . . 6 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 (𝐹𝑥) = 𝑦)
7 eqcom 2826 . . . . . . 7 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
87reubii 3390 . . . . . 6 (∃!𝑥𝐴 (𝐹𝑥) = 𝑦 ↔ ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
96, 8sylib 220 . . . . 5 ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = (𝐹𝑥))
10 disjrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
1110eleq2d 2896 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
124, 9, 11rmoxfrd 30249 . . . 4 (𝜑 → (∃*𝑦𝐶 𝑧𝐷 ↔ ∃*𝑥𝐴 𝑧𝐵))
1312bicomd 225 . . 3 (𝜑 → (∃*𝑥𝐴 𝑧𝐵 ↔ ∃*𝑦𝐶 𝑧𝐷))
1413albidv 1915 . 2 (𝜑 → (∀𝑧∃*𝑥𝐴 𝑧𝐵 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷))
15 df-disj 5023 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑧∃*𝑥𝐴 𝑧𝐵)
16 df-disj 5023 . 2 (Disj 𝑦𝐶 𝐷 ↔ ∀𝑧∃*𝑦𝐶 𝑧𝐷)
1714, 15, 163bitr4g 316 1 (𝜑 → (Disj 𝑥𝐴 𝐵Disj 𝑦𝐶 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wal 1529   = wceq 1531  wcel 2108  ∃!wreu 3138  ∃*wrmo 3139  Disj wdisj 5022  wf 6344  1-1-ontowf1o 6347  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-disj 5023  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  tocyccntz  30779  volmeas  31483  carsggect  31569
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