Theorem imasetpreimafvbijlemfv 47766

Description: Lemma for imasetpreimafvbij 47770: 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 6600	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
2	1	anim1i 616	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃) → (Fun 𝐹 ∧ 𝑌 ∈ 𝑃))
3	2	3adant3 1133	. . 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 47764	. . 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 1134	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → Fun 𝐹)
9		funiunfv 7204	. . 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 1139	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → 𝑋 ∈ 𝑌)
12		simpl1 1193	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝐹 Fn 𝐴)
13		simpl2 1194	. . . . 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 1195	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → 𝑋 ∈ 𝑌)
16	4	elsetpreimafveqfv 47756	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑌 ∈ 𝑃 ∧ 𝑦 ∈ 𝑌 ∧ 𝑋 ∈ 𝑌)) → (𝐹‘𝑦) = (𝐹‘𝑋))
17	12, 13, 14, 15, 16	syl13anc 1375	. . . 4 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) ∧ 𝑦 ∈ 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑋))
18	17	ralrimiva 3130	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
19		fveq2 6842	. . . 4 ⊢ (𝑦 = 𝑋 → (𝐹‘𝑦) = (𝐹‘𝑋))
20	19	iuneqconst 4960	. . 3 ⊢ ((𝑋 ∈ 𝑌 ∧ ∀𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋)) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
21	11, 18, 20	syl2anc 585	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → ∪ 𝑦 ∈ 𝑌 (𝐹‘𝑦) = (𝐹‘𝑋))
22	7, 10, 21	3eqtr2d 2778	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝑃 ∧ 𝑋 ∈ 𝑌) → (𝐻‘𝑌) = (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  {csn 4582  ∪ cuni 4865  ∪ ciun 4948   ↦ cmpt 5181  ^◡ccnv 5631   “ cima 5635  Fun wfun 6494   Fn wfn 6495  ‘cfv 6500

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  imasetpreimafvbijlemfv1  47767