Theorem disjrdx 33067

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 6818	. . . . . . 7 ⊢ (𝐹:𝐴–1-1-onto→𝐶 → 𝐹:𝐴⟶𝐶)
3	1, 2	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffvelcdmda 7078	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶)
5		f1ofveu 7408	. . . . . . 7 ⊢ ((𝐹:𝐴–1-1-onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
6	1, 5	sylan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)
7		eqcom 2767	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
8	7	reubii 3374	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 ↔ ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
9	6, 8	sylib 221	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃!𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
10		disjrdx.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)
11	10	eleq2d 2846	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵))
12	4, 9, 11	rmoxfrd 32971	. . . 4 ⊢ (𝜑 → (∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
13	12	bicomd 226	. . 3 ⊢ (𝜑 → (∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
14	13	albidv 1953	. 2 ⊢ (𝜑 → (∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
15		df-disj 5071	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑧∃*𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16		df-disj 5071	. 2 ⊢ (Disj 𝑦 ∈ 𝐶 𝐷 ↔ ∀𝑧∃*𝑦 ∈ 𝐶 𝑧 ∈ 𝐷)
17	14, 15, 16	3bitr4g 317	1 ⊢ (𝜑 → (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑦 ∈ 𝐶 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃!wreu 3363  ∃*wrmo 3364  Disj wdisj 5070  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-disj 5071  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541

This theorem is used by:  tocyccntz  33587  volmeas  34745  carsggect  34832