Theorem uniimaprimaeqfv 48667

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 6741	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
2	1	birani 509	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴⟶ran 𝐹)
3		cnvimass 6210	. . . . 5 ⊢ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ dom 𝐹
4		fndm 6661	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 3955	. . . 4 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
6	5	adantr 486	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
7		preimafvsnel 48664	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
8	2, 6, 7	3jca 1146	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})))
9		fniniseg 7080	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
10	9	adantr 486	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
11		simpr 490	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋)) → (𝐹‘𝑥) = (𝐹‘𝑋))
12	10, 11	biimtrdi 256	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) → (𝐹‘𝑥) = (𝐹‘𝑋)))
13	12	ralrimiv 3157	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋))
14		uniimafveqt 48666	. 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)))
15	8, 13, 14	sylc 66	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  {csn 4588  ∪ cuni 4870  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658   Fn wfn 6550  ⟶wf 6551  ‘cfv 6555

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  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-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563

This theorem is used by:  imasetpreimafvbijlemfo  48690  fundcmpsurbijinjpreimafv  48692