Theorem uniimaprimaeqfv 47842

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 6680	. . . . 5 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
2	1	biimpi 216	. . . 4 ⊢ (𝐹 Fn 𝐴 → 𝐹:𝐴⟶ran 𝐹)
3	2	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴⟶ran 𝐹)
4		cnvimass 6047	. . . . 5 ⊢ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ dom 𝐹
5		fndm 6601	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	4, 5	sseqtrid 3964	. . . 4 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
7	6	adantr 480	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
8		preimafvsnel 47839	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
9	3, 7, 8	3jca 1129	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})))
10		fniniseg 7012	. . . . 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	biimtrdi 253	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) → (𝐹‘𝑥) = (𝐹‘𝑋)))
14	13	ralrimiv 3128	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋))
15		uniimafveqt 47841	. 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889  {csn 4567  ∪ cuni 4850  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634   Fn wfn 6493  ⟶wf 6494  ‘cfv 6498

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506

This theorem is referenced by:  imasetpreimafvbijlemfo  47865  fundcmpsurbijinjpreimafv  47867