Theorem afv2elrn 43437

Description: An alternate function value belongs to the range of the function, analogous to fvelrn 6846. (Contributed by AV, 3-Sep-2022.)

Assertion

Ref	Expression
afv2elrn	⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)

Proof of Theorem afv2elrn

Step	Hyp	Ref	Expression
1		fundmdfat 43335	. 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
2		dfatafv2rnb 43433	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
3	1, 2	sylib 220	1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114  dom cdm 5557  ran crn 5558  Fun wfun 6351   defAt wdfat 43322  ''''cafv2 43414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-nel 3126  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-dfat 43325  df-afv2 43415

This theorem is referenced by:  fnafv2elrn  43439