Theorem iuneq1 5013

Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.)

Assertion

Ref	Expression
iuneq1	⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iuneq1

Step	Hyp	Ref	Expression
1		iunss1 5011	. . 3 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶)
2		iunss1 5011	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)
3	1, 2	anim12i 613	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶))
4		eqss 4011	. 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
5		eqss 4011	. 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶))
6	3, 4, 5	3imtr4i 292	1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ⊆ wss 3963  ∪ ciun 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-v 3480  df-ss 3980  df-iun 4998

This theorem is referenced by:  iuneq1d  5024  iinvdif  5085  iunxprg  5101  iununi  5104  iunsuc  6471  funopsn  7168  funiunfv  7268  onfununi  8380  iunfi  9381  ttrclselem1  9763  ttrclselem2  9764  rankuni2b  9891  pwsdompw  10241  ackbij1lem7  10263  r1om  10281  fictb  10282  cfsmolem  10308  ituniiun  10460  domtriomlem  10480  domtriom  10481  inar1  10813  fsum2d  15804  fsumiun  15854  ackbijnn  15861  fprod2d  16014  prmreclem5  16954  lpival  21352  fiuncmp  23428  ovolfiniun  25550  ovoliunnul  25556  finiunmbl  25593  volfiniun  25596  voliunlem1  25599  iuninc  32581  ofpreima2  32683  gsumpart  33043  esum2dlem  34073  sigaclfu2  34102  sigapildsyslem  34142  fiunelros  34155  bnj548  34890  bnj554  34892  bnj594  34905  neibastop2lem  36343  istotbnd3  37758  0totbnd  37760  sstotbnd2  37761  sstotbnd  37762  sstotbnd3  37763  totbndbnd  37776  prdstotbnd  37781  cntotbnd  37783  heibor  37808  dfrcl4  43666  iunrelexp0  43692  comptiunov2i  43696  corclrcl  43697  cotrcltrcl  43715  trclfvdecomr  43718  dfrtrcl4  43728  corcltrcl  43729  cotrclrcl  43732  fiiuncl  45005  sge0iunmptlemfi  46369  caragenfiiuncl  46471  carageniuncllem1  46477  ovnsubadd2lem  46601