Theorem infxpidmlem6 4938

Description: Lemma for infxpidm 4945. A simple but frequently used fact.

Hypotheses

Ref	Expression
infxpidmlem.1	⊢ H = {f∣(f = ∅ ∨ ∃t((ω ≼ t ∧ t ⊆ A) ∧ f:(t × t)–1-1-onto→t))}
infxpidmlem6.2	⊢ B = ran ∪C

Assertion

Ref	Expression
infxpidmlem6	⊢ (y ∈ B ↔ ∃g ∈ C y ∈ ran g)

Distinct variable group(s):   y,f,g,t,A   y,B,f,g,t   y,C,f,g,t   y,H,g

Proof of Theorem infxpidmlem6

Step	Hyp	Ref	Expression
1		infxpidmlem6.2	. . . 4 ⊢ B = ran ∪C
2		rnuni 2646	. . . 4 ⊢ ran ∪C = ∪g ∈ C ran g
3	1, 2	eqtr 1119	. . 3 ⊢ B = ∪g ∈ C ran g
4	3	eleq2i 1153	. 2 ⊢ (y ∈ B ↔ y ∈ ∪g ∈ C ran g)
5		eliun 1998	. 2 ⊢ (y ∈ ∪g ∈ C ran g ↔ ∃g ∈ C y ∈ ran g)
6	4, 5	bitr 151	1 ⊢ (y ∈ B ↔ ∃g ∈ C y ∈ ran g)

Syntax hints:   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919  ∪ciun 1994   class class class wbr 2054  ωcom 2372   × cxp 2408  ran crn 2411  –1-1-onto→wf1o 2421   ≼ cdom 3272

This theorem is referenced by:  infxpidmlem7 4939  infxpidmlem8 4940

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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