Theorem iunpreima 32918

Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)

Assertion

Ref	Expression
iunpreima	⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iunpreima

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4960	. . . . 5 ⊢ ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵)
2	1	a1i 11	. . . 4 ⊢ (Fun 𝐹 → ((𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵))
3	2	rabbidv 3423	. . 3 ⊢ (Fun 𝐹 → {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵})
4		funfn 6566	. . . 4 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
5		fncnvima2 7056	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
6	4, 5	sylbi 220	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ ∪ 𝑥 ∈ 𝐴 𝐵})
7		iunrab 5017	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵}
8	7	a1i 11	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵} = {𝑦 ∈ dom 𝐹 ∣ ∃𝑥 ∈ 𝐴 (𝐹‘𝑦) ∈ 𝐵})
9	3, 6, 8	3eqtr4d 2808	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
10		fncnvima2 7056	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
11	4, 10	sylbi 220	. . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ 𝐵) = {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
12	11	iuneq2d 4987	. 2 ⊢ (Fun 𝐹 → ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵) = ∪ 𝑥 ∈ 𝐴 {𝑦 ∈ dom 𝐹 ∣ (𝐹‘𝑦) ∈ 𝐵})
13	9, 12	eqtr4d 2801	1 ⊢ (Fun 𝐹 → (◡𝐹 “ ∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  {crab 3416  ∪ ciun 4956  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by:  elrspunidl  33745