Theorem disjrdx 32791

Description: Re-index a disjunct collection statement. (Contributed by Thierry Arnoux, 7-Apr-2017.)

Hypotheses

Ref	Expression
disjrdx.1	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
disjrdx.2	⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)

Assertion

Ref	Expression
disjrdx	⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 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.

Step	Hyp	Ref	Expression
1		disjrdx.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)
2		f1of 6806	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴⟶𝐶)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffvelcdmda 7065	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶)
5		f1ofveu 7390	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
6	1, 5	sylan 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7		eqcom 2769	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
8	7	reubii 3376	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
9	6, 8	sylib 220	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
10		disjrdx.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)
11	10	eleq2d 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵))
12	4, 9, 11	rmoxfrd 32692	. . . 4 ⊢ (𝜑 → (∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
13	12	bicomd 225	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
14	13	albidv 1940	. 2 ⊢ (𝜑 → (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
15		df-disj 5068	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16		df-disj 5068	. 2 ⊢ (Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)
17	14, 15, 16	3bitr4g 316	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ∃!wreu 3365  ∃*wrmo 3366  Disj wdisj 5067  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-disj 5068  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529

This theorem is referenced by:  tocyccntz  33324  volmeas  34528  carsggect  34615