Theorem iunrdx 32586

Description: Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)

Hypotheses

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

Assertion

Ref	Expression
iunrdx	⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 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 iunrdx

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunrdx.1	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴–onto→𝐶)
2		fof 6834	. . . . . . 7 ⊢ (𝐹:𝐴–onto→𝐶 → 𝐹:𝐴⟶𝐶)
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)
4	3	ffvelcdmda 7118	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐶)
5		foelrn 7141	. . . . . 6 ⊢ ((𝐹:𝐴–onto→𝐶 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
6	1, 5	sylan 579	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
7		iunrdx.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → 𝐷 = 𝐵)
8	7	eleq2d 2830	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 = (𝐹‘𝑥)) → (𝑧 ∈ 𝐷 ↔ 𝑧 ∈ 𝐵))
9	4, 6, 8	rexxfrd 5427	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
10	9	bicomd 223	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷))
11	10	abbidv 2811	. 2 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷})
12		df-iun 5017	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
13		df-iun 5017	. 2 ⊢ ∪ 𝑦 ∈ 𝐶 𝐷 = {𝑧 ∣ ∃𝑦 ∈ 𝐶 𝑧 ∈ 𝐷}
14	11, 12, 13	3eqtr4g 2805	1 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐶 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∃wrex 3076  ∪ ciun 5015  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581

This theorem is referenced by:  volmeas  34195