Theorem cnvimassrndm 6085

Description: The preimage of a superset of the range of a class is equal to the domain of the class. Generalization of cnvimarndm 6086 to supersets of the range. (Contributed by AV, 18-Sep-2024.) (Proof shortened by BJ, 29-Sep-2026.)

Assertion

Ref	Expression
cnvimassrndm	⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

Proof of Theorem cnvimassrndm

Step	Hyp	Ref	Expression
1		dfdm4 5882	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		df-rn 5666	. . . 4 ⊢ ran 𝐹 = dom ◡𝐹
3	2	sseq1i 3941	. . 3 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ dom ◡𝐹 ⊆ 𝐴)
4		dfrn7 6072	. . 3 ⊢ (dom ◡𝐹 ⊆ 𝐴 → ran ◡𝐹 = (◡𝐹 “ 𝐴))
5	3, 4	sylbi 220	. 2 ⊢ (ran 𝐹 ⊆ 𝐴 → ran ◡𝐹 = (◡𝐹 “ 𝐴))
6	1, 5	eqtr2id 2811	1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3900  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  cnvimarndm  6086  fnco  6668  fimacnv  6743