Theorem imasetpreimafvbijlemfv 47512

Description: Lemma for imasetpreimafvbij 47516: 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 6581	. . . . 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 47510	. . 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 7182	. . 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 47502	. . . . 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 3124	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
19		fveq2 6822	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
20	19	iuneqconst 4951	. . 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 2772	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  {csn 4573  ∪ cuni 4856  ∪ ciun 4939   ↦ cmpt 5170  ^◡ccnv 5613   “ cima 5617  Fun wfun 6475   Fn wfn 6476  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  imasetpreimafvbijlemfv1  47513