Theorem bdceqir 16543

Description: A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 16542) equality in the hypothesis, to work better with definitions (𝐵 is the definiendum that one wants to prove bounded; see comment of bd0r 16524). (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdceqir.min	⊢ BOUNDED 𝐴
bdceqir.maj	⊢ 𝐵 = 𝐴

Assertion

Ref	Expression
bdceqir	⊢ BOUNDED 𝐵

Proof of Theorem bdceqir

Step	Hyp	Ref	Expression
1		bdceqir.min	. 2 ⊢ BOUNDED 𝐴
2		bdceqir.maj	. . 3 ⊢ 𝐵 = 𝐴
3	2	eqcomi 2235	. 2 ⊢ 𝐴 = 𝐵
4	1, 3	bdceqi 16542	1 ⊢ BOUNDED 𝐵

Syntax hints:   = wceq 1398  BOUNDED wbdc 16539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2213  ax-bd0 16512

This theorem depends on definitions:  df-bi 117  df-cleq 2224  df-clel 2227  df-bdc 16540

This theorem is referenced by:  bdcrab  16551  bdccsb  16559  bdcdif  16560  bdcun  16561  bdcin  16562  bdcnulALT  16565  bdcpw  16568  bdcsn  16569  bdcpr  16570  bdctp  16571  bdcuni  16575  bdcint  16576  bdciun  16577  bdciin  16578  bdcsuc  16579  bdcriota  16582