Theorem cnvimassrndm 6079

Description: The preimage of a superset of the range of a class is equal to the domain of the class. Generalization of cnvimarndm 6080 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 5877	. 2 ⊢ dom 𝐹 = ran ◡𝐹
2		df-rn 5662	. . . 4 ⊢ ran 𝐹 = dom ◡𝐹
3	2	sseq1i 3959	. . 3 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ dom ◡𝐹 ⊆ 𝐴)
4		dfrn7 6066	. . 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 2809	1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  cnvimarndm  6080  fnco  6657  fimacnv  6732