Theorem uniimaprimaeqfv 44722

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 6597	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
2	1	biimpi 215	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
3	2	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴⟶ran 𝐹)
4		cnvimass 5978	. . . . 5 ⊢ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ dom 𝐹
5		fndm 6520	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	sseqtrid 3969	. . . 4 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
7	6	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
8		preimafvsnel 44719	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
9	3, 7, 8	3jca 1126	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})))
10		fniniseg 6919	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
11	10	adantr 480	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
12		simpr 484	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋)) → (𝐹‘𝑥) = (𝐹‘𝑋))
13	11, 12	syl6bi 252	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) → (𝐹‘𝑥) = (𝐹‘𝑋)))
14	13	ralrimiv 3106	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋))
15		uniimafveqt 44721	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ⊆ wss 3883  {csn 4558  ∪ cuni 4836  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   “ cima 5583   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426

This theorem is referenced by:  imasetpreimafvbijlemfo  44745  fundcmpsurbijinjpreimafv  44747