Theorem imasetpreimafvbijlemfv 47407

Description: Lemma for imasetpreimafvbij 47411: the value of the mapping 𝐻 at a preimage of a value of function 𝐹. (Contributed by AV, 5-Mar-2024.)

Hypotheses

Ref	Expression
fundcmpsurinj.p	⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
fundcmpsurinj.h	⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))

Assertion

Ref	Expression
imasetpreimafvbijlemfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Distinct variable groups:   𝑥,𝐴,𝑧   𝑥,𝐹,𝑧,𝑝   𝑃,𝑝   𝑋,𝑝   𝐴,𝑝,𝑥,𝑧   𝑥,𝑃   𝑥,𝑋   𝑌,𝑝,𝑥,𝑧

Allowed substitution hints:   𝑃(𝑧)   𝐻(𝑥,𝑧,𝑝)   𝑋(𝑧)

Proof of Theorem imasetpreimafvbijlemfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnfun 6621	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	anim1i 615	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
3	2	3adant3 1132	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
4		fundcmpsurinj.p	. . . 4 ⊢ 𝑃 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (◡𝐹 “ {(𝐹‘𝑥)})}
5		fundcmpsurinj.h	. . . 4 ⊢ 𝐻 = (𝑝 ∈ 𝑃 ↦ ∪ (𝐹 “ 𝑝))
6	4, 5	fundcmpsurinjlem3 47405	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ 𝑃) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
7	3, 6	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = ∪ (𝐹 “ 𝑌))
8	1	3ad2ant1 1133	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → Fun 𝐹)
9		funiunfv 7225	. . 3 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
10	8, 9	syl 17	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = ∪ (𝐹 “ 𝑌))
11		simp3 1138	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
12		simpl1 1192	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝐹 Fn 𝐴)
13		simpl2 1193	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑌 ∈ 𝑃)
14		simpr 484	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
15		simpl3 1194	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑋 ∈ 𝑌)
16	4	elsetpreimafveqfv 47397	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑌 ∈ 𝑃 ∧ 𝑦 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑋))
17	12, 13, 14, 15, 16	syl13anc 1374	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑋))
18	17	ralrimiva 3126	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
19		fveq2 6861	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
20	19	iuneqconst 4970	. . 3 ⊢ ((𝑋 ∈ 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
21	11, 18, 20	syl2anc 584	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
22	7, 10, 21	3eqtr2d 2771	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  {csn 4592  ∪ cuni 4874  ∪ ciun 4958   ↦ cmpt 5191  ^◡ccnv 5640   “ cima 5644  Fun wfun 6508   Fn wfn 6509  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  imasetpreimafvbijlemfv1  47408