Theorem disjrdx 28786

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 6050	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴⟶𝐶)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffvelrnda 6267	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶)
5		f1ofveu 6544	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
6	1, 5	sylan 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7		eqcom 2617	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
8	7	reubii 3105	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
9	6, 8	sylib 207	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
10		disjrdx.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)
11	10	eleq2d 2673	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵))
12	4, 9, 11	rmoxfrd 28717	. . . 4 ⊢ (𝜑 → (∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
13	12	bicomd 212	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
14	13	albidv 1836	. 2 ⊢ (𝜑 → (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
15		df-disj 4554	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16		df-disj 4554	. 2 ⊢ (Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)
17	14, 15, 16	3bitr4g 302	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃!wreu 2898  ∃*wrmo 2899  Disj wdisj 4553  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-disj 4554  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  volmeas  29621  carsggect  29707