Theorem uniimaprimaeqfv 44295

Description: The union of the image of the preimage of a function value is the function value. (Contributed by AV, 12-Mar-2024.)

Assertion

Ref	Expression
uniimaprimaeqfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))

Proof of Theorem uniimaprimaeqfv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn3 6514	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
2	1	biimpi 219	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
3	2	adantr 484	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴⟶ran 𝐹)
4		cnvimass 5925	. . . . 5 ⊢ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ dom 𝐹
5		fndm 6440	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	sseqtrid 3946	. . . 4 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
7	6	adantr 484	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
8		preimafvsnel 44292	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
9	3, 7, 8	3jca 1125	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})))
10		fniniseg 6825	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
11	10	adantr 484	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
12		simpr 488	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋)) → (𝐹‘𝑥) = (𝐹‘𝑋))
13	11, 12	syl6bi 256	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) → (𝐹‘𝑥) = (𝐹‘𝑋)))
14	13	ralrimiv 3112	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋))
15		uniimafveqt 44294	. 2 ⊢ ((𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})) → (∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋)))
16	9, 14, 15	sylc 65	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860  {csn 4525  ∪ cuni 4801  ^◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347

This theorem is referenced by:  imasetpreimafvbijlemfo  44318  fundcmpsurbijinjpreimafv  44320