Theorem iunxsng 4045

Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.)

Hypothesis

Ref	Expression
iunxsng.1	⊢ (x = A → B = C)

Assertion

Ref	Expression
iunxsng	⊢ (A ∈ V → ∪x ∈ {A}B = C)

Distinct variable groups:   x,A   x,C

Allowed substitution hints:   B(x)   V(x)

Proof of Theorem iunxsng

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ {A}B ↔ ∃x ∈ {A}y ∈ B)
2		iunxsng.1	. . . . 5 ⊢ (x = A → B = C)
3	2	eleq2d 2420	. . . 4 ⊢ (x = A → (y ∈ B ↔ y ∈ C))
4	3	rexsng 3767	. . 3 ⊢ (A ∈ V → (∃x ∈ {A}y ∈ B ↔ y ∈ C))
5	1, 4	syl5bb 248	. 2 ⊢ (A ∈ V → (y ∈ ∪x ∈ {A}B ↔ y ∈ C))
6	5	eqrdv 2351	1 ⊢ (A ∈ V → ∪x ∈ {A}B = C)

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {csn 3738  ∪ciun 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-sn 3742  df-iun 3972

This theorem is used by:  iunxsn  4046