Theorem uniimaprimaeqfv 47993

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 6706	. . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
2	1	birani 507	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹:𝐴⟶ran 𝐹)
3		cnvimass 6073	. . . . 5 ⊢ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ dom 𝐹
4		fndm 6626	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrid 3980	. . . 4 ⊢ (𝐹 Fn 𝐴 → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
6	5	adantr 484	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴)
7		preimafvsnel 47990	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)}))
8	2, 6, 7	3jca 1142	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹:𝐴⟶ran 𝐹 ∧ (◡𝐹 “ {(𝐹‘𝑋)}) ⊆ 𝐴 ∧ 𝑋 ∈ (◡𝐹 “ {(𝐹‘𝑋)})))
9		fniniseg 7043	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
10	9	adantr 484	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) ↔ (𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋))))
11		simpr 488	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝐹‘𝑥) = (𝐹‘𝑋)) → (𝐹‘𝑥) = (𝐹‘𝑋))
12	10, 11	biimtrdi 255	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)}) → (𝐹‘𝑥) = (𝐹‘𝑋)))
13	12	ralrimiv 3155	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ (◡𝐹 “ {(𝐹‘𝑋)})(𝐹‘𝑥) = (𝐹‘𝑋))
14		uniimafveqt 47992	. 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 65	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ∪ (𝐹 “ (◡𝐹 “ {(𝐹‘𝑋)})) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906  {csn 4584  ∪ cuni 4867  ^◡ccnv 5648  dom cdm 5649  ran crn 5650   “ cima 5652   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531

This theorem is referenced by:  imasetpreimafvbijlemfo  48016  fundcmpsurbijinjpreimafv  48018